1. REAL NUMBERS - TopperLearning

www.plancess.com

1. REAL NUMBERS

1. IntroductionIs it possible to think of a life without the numbers? Let us look at a few scenarios –

1. For measuring the speed of a bike / car – we need numbers

2. For understanding whether one has fever or not – we need numbers

3. For understanding how high the Himalayas are – we need numbers

4. For understanding the time – we need numbers

5. To pay for our shopping – we need numbers

6. To know how much money we have in our purse – we need numbers

7. To have a measure of our weight – we need numbers

8. Even computer understands only two symbols - 0 and 1 (every command is understood by the computers using 0’s and 1’s).

So if we think about it, it seems like, we are surrounded by numbers and our lives would come to stand still without these numbers.

L. Kronecker (1823-1891), the reputed German Mathematician, said that “God gave us the natural number, all else is the work of man”. �is statement also reveals in a nut shell the signi�cant role of the universe of numbers played in the evolution of human thought.

N: �e set of natural numbers,

W: �e set of whole numbers,

Z: �e set of Integers,

Q: �e set of rationals,

R: �e set of Real Numbers.

N is a subset of W, which is a subset of Z, which in turn is a subset of Q, which is a subset of R.

Figure 1.1

1.2 Real Numbers

www.plancess.com

2. Classification of Numbers

2.1 Natural Numbers (N)Set of all counting numbers from 1 to ,N 1,2,3,4,... .=

2.1.1 Composite Numbers

All natural numbers (except 1), which are not prime are Composite Numbers. A number which is divisible by other numbers except one and itself. If C is the set of composite numbers then, C = {4, 6, 8, 9, 10 ...}.

Note: 1 is neither prime nor composite number.

2.1.2 Prime Numbers

A number which is not divisible by any natural number except unity and itself is known as Prime number.

e.g., 2, 3, 5, 7, 11, 13, 17, 19, 23....etc. If P is the set of prime numbers, then P = {2, 3, 5, 7......}.

Prime numbers are divided into three types:

(i) Co-prime (Relatively prime) (ii) Twin primes (iii) Prime Triplet

Co-prime Numbers: If the Highest Common Factor of the given numbers (not necessarily prime) is 1 then they are called Co-prime Numbers. e.g., 4, 9 are co-prime as H.C.F. of (4, 9) = 1.

Note: - Any two consecutive numbers will always be co-prime.

2.2 Whole Numbers (W)Set of all natural numbers including zero, W 0,1,2,3, 4, ... .=

1.3 Foundation for Mathematics

www.plancess.com

2.3 Integers (Z or I)Set of all positive and negative counting numbers including zero from to

I or Z = {...,-3, -2, -1, 0, 1, 2, 3 ... …}

Positive integers = {1, 2, 3, 4 … …}

Negative integers = {… …. …. –3, –2, –1}

2.3.1 Even Numbers

All integers which are divisible by 2 are called even numbers. Even numbers are denoted by the expression 2n, where n is any integer, so, if E is a set of even numbers, then E = {.......-6, -4, -2, 0, 2, 4, 6........}.

2.3.2 Odd Numbers

All integers which are not divisible by 2 are called odd numbers. Odd numbers are denoted by the general expression 2n - 1 where n is any integer. If O is a set of odd numbers, then

O = {..., -7, -5, -3, -1, 1, 3, 5, 7........}.

2.4 Rational Numbers (Q)�ese are real numbers which can be expressed in the form of p/q, where p and q are integers and q 0. e.g., 2/3, 37/15, -17/19.

2.4.1 Rational Numbers in Decimal Form

Every rational number when expressed in decimal form is expressible either in Terminating or in Non-Terminating Repeating Decimal form.

2.4.2 Classification of Decimals

(a) Terminating Decimals: Rational numbers pq

for which long division terminates after a �nite

number of steps are called Terminating Decimals.

e.g., (0.75, – 0.02, 3350

etc.)

(b) Non-Terminating Repeating (Recurring) Decimals Numbers: Rational numbers pq

for which the long division is non-terminating but repeating are called Non Terminating or Recurring Decimals

e.g., (0.666, –2.333, 1790

etc.)

2.4.3 Fraction

(a) Common Fraction: Fraction whose denominator is not 10 is called Common Fraction.

(b) Decimal Fraction: Fraction whose denominator is 10 or any power of 10 is called Decimal Fraction.

1.4 Real Numbers

www.plancess.com

(c) Proper Fraction: Fraction in which Numerator is less than the Denominator i.e., 23

is called Proper Fraction.

(d) Improper fraction: Fraction in which Numerator is greater than the Denominator i.e., 74

is called Improper fraction.

(e) Equivalent Fractions: Two or more functions said to be equivalent if the multiplication or division of both the numerator as well as the denominator of �rst fraction by the same number gives another fraction.

e.g. 9/5 is multiplied by 3 for numerator as well as denominator, (9×3)/(5×3) = 27 / 15

Here, 9 / 5 and 27 / 15 are called equivalent fraction.

(f ) Like Fractions: Fractions having the same denominators are called the like fractions.

e.g. 1/5, 3/5, 7/5 ... etc.

(g) Unlike Fractions: Fractions having di�erent denominators are called unlike fractions.

e.g. 2/5, 4/3, 7/9 ... etc.

(h) Mixed fraction: Fraction which combines whole number and fractional part i.e., 4 37

is called mixed fraction.

(i) Complex fraction: Fraction in which both Numerator and Denominator are fraction i.e., 4 / 56 / 7is called Complex fraction.

(j) Compound fraction: Fraction of a fraction is called compound fraction i.e. 2 4 83 5 15

=

(k) Continued fraction: Fraction consists of the fractional denominators

i.e., 11

12

13

14

5

Note:- Improper fraction can be written in the form of mixed fraction.

2.5 Irrational Numbers (Non-Terminating-Non-Repeating Decimals)Numbers which can neither be expressed as terminating decimal nor as a repeating decimal is called Irrational Numbers.

e.g. 43 72 , 4 ,2 3 , 2 3 , 3 etc.

2.6 Real Numbers (R)Rational numbers and non-rational numbers taken together are called Real Numbers. �us real numbers is either rational or irrational but cannot be simultaneously both. Numbers which can represent actual


www.plancess.com

physical quantities in a meaningful way. �ese can be represented on the number line. Number line is geometrical straight line with arbitrarily de�ned zero (origin).

PLANCESS CONCEPTS

Absolute value of a real number: �e absolute value or modulus of a real number x, denoted

by |x|, is de�ned as |x| = x, if x 0–x, if x 0

Problem Solving Trick – In many questions, the absolute value function is used. In such cases, the solution is divided in two parts, one for x ≥ 0 and another for x < 0. �en according to the de�nition of |x|, we substitute x or –x accordingly in place of |x| for the two cases and then solve the question.

Shivam Agarwal

Gold Medalist, INPhO

Illustration 1: Prove that 3 3 is irrational.

Solution: Let 3 3 be rational = pq

, where p and q Z and p, q have no common factor except 1

also q > 1.

∴ pq

= 3 3 and on cubing both sides we have 3

3

p

q = 3

Multiply both sides by q2,3

2p3q

q= . Clearly L.H.S is rational since p, q have no common factor

therefore p3, q also have no common factor while R.H.S. is an integer.

∴ L.H.S R.H.S which contradicts our assumption that 3 3 is rational.

∴ 3 3 is irrational.

Illustration 2: Prove that 2 + 3 is irrational.

Solution: Let 2 + 3 be a rational number equal to r

∴ 2 + 3 = r ⇒ 3 = r – 2

Here L.H.S is an irrational number while R.H.S. r – 2 is rational. ∴ L.H.S R.H.S

Hence it contradicts our assumption that 2 + 3 is rational.

∴ 2 + 3 is irrational.

Illustration 3: Prove that 2 3 is irrational.

Solution: Let 2 3 be rational number say ‘x’, i.e. x = 2 3

⇒ x2 = 2 + 3 + 2 3 2 = 5 + 62

1.6 Real Numbers

www.plancess.com

⇒ x2 = 5 + 62 ⇒ 6 = 2x 52

As x, 5 and 2 are rationals ⇒ 2x 52

is a rational number.

2x 56

2⇒ = is a rational number

We know that 6 is a irrational number, hence our assumption is wrong ⇒ 2 3 is an irrational number.

3. Euclid’s Division Lemma or Euclid’s Division AlgorithmFor any two positive integers a and b there exist unique integers q and r such that

a = bq + r, where 0 r < b.

Let us consider a = 217, b = 5 and make the division of 217 by 5 as under:

Dividend

Divisor 5) 217 (43 Quotient

20

17

15

2 Remainder

i.e. Dividend = Divisor × Quotient + Remainder

(a) = (b) × (q) + (r)

For e.g. consider positive integers 18 and 4

18 = 4 × 4 + 2

For 18 (= a) and 4 (= b) we have q = 4, r = 2 and 0 r < b

In the relation a = bq + r, where 0 r < b is nothing but a statement of the long division of number a by b in which q is the quotient obtained and r is the remainder.

Illustration 4: Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m +1 for some integer m.

Solution: Let a and b are two positive integers such that a is greater than b, then:

a = bq + r; where q and r are also positive integers and 0 r < b


www.plancess.com

Taking b = 3, we get:

a = 3q + r; where 0 r < 3

⇒ the value of positive integer a will be 3q + 0, 3q + 1 or 3q + 2

i.e., 3q, 3q + 1 or 3q + 2.

Now we have to show that the square of positive integers 3q, 3q + 1 and 3q + 2 can be expressed as

3m or 3m + 1 for some integer m.

∴ Square of 3q = (3q)2 = 9q2 = 3(3q2) = 3m

Where m is some integer and m = 3q2

Square of 3q + 1 = (3q + 1)2 = 9q2 + 6q + 1 = 3(3q2 + 2q) + 1

= 3m + 1 where m is some integer and m = 3q2 + 2q.

Square of 3q + 2 = (3q + 2)2= 9q2 + 12q + 4

= 9q2 + 12q + 3 + 1 = 3(3q2 + 4q + 1) + 1 = 3m + 1 for some integer and m

∴ �e square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

3.1 Application of Euclid’s Division Lemma: Finding H.C.F.Consider positive integers 418 and 33

Step 1: Taking bigger number (418) as a and smaller number (33) as b.

Express the numbers as a = bq + r; 418 = 33 × 12 + 22

Step 2: Now taking the divisor 33 and remainder 22, apply the Euclid’s division method to get.

33 = 22 × 1 + 11 [Expressing as a = bq + r]

Step 3: Again with new division 22 and new remainder 11, apply the Euclid’s division algorithm to get 22 = 11 × 2 + 0

Step 4: Repeat until the remainder = 0 (as we cannot proceed further).

Step 5: �e last divisor is 11 and we say H.C.F. of 418 and 33 = 11

Illustration 5: Find the HCF of 1848, 3058 and 1331.

Solution: Let us take two bigger numbers 1848 and 3058, where 3058 > 1848

3058 = 1848 × 1 + 1210

1848 = 1210 × 1 + 638 [Using Euclid’s division lemma]

1210 = 638 × 1 + 572

638 = 572 × 1 + 66

1.8 Real Numbers

www.plancess.com

572 = 66 × 8 + 44

66 = 44 × 1 + 22

44 = 22 × 2 + 0

∴ HCF (1848, 3058) = 22.

Let us now �nd the HCF of the numbers 1331 and 22.

1331 = 22 × 60 + 11 22 = 11 × 2 + 0

∴ HCF (22, 1331) = 11

Hence the HCF of the three given numbers 1848, 3058 and 1331 is 11.

PLANCESS CONCEPTS

Relation between HCF and LCM : For any two numbers a and b,

(a) L.C.M. (a, b) = Product of each prime factor with highest powers a bH.C.F. (a, b)

=

(b) H.C.F. (a, b) = Product of common prime factor with lowest powers a bL .C. M. (a, b)

=

Remark: �e above relations hold only for two numbers.

Shivam Agarwal


4. Divisibility of Numbers and Algebraic Expressions4.1 Divisibility Tests of Numbers

No. Divisibility Test Examples

2 Unit digit should be even. 4096, 23548 as they end with 6 and 8 i.e., even numbers

3 �e sum of digits of no. should be divisible by 3

2143251, sum of the digits is 18 and it is divisible by 3

4 �e number formed by last 2 digits of given no. should be divisible by 4.

548, here 48 4 = 12 and it is divisible by 4

5 Unit digit should be 0 or 5. 4095 and 235060 as they have 5, 0 at unit places.

6 Number should be divisible by 2 and 3 both.

753618, sum of the digits is 30 and it is divisible by 2 and 3.


www.plancess.com

No. Divisibility Test Examples


5432, here 432 is divisible by 8

9 Sum of digits of given no. should be divisible by 9

125847, sum of the digits is 27 and it is divisible by 9

11 �e di�erence between sums of the digits at even and at odd places should be zero or multiple of 11.

9582540, here sum of odd places- sum of even places (22 – 11 = 11) and 11 is a divisible by 11.

25 Last 2 digits of the number should be 00, 25, 50 or 75.

2500, 2550 etc.

Rule for 7: Double the last digit of given number and subtract it from the remaining number. �e result should be zero or divisible by 7.

e.g., 6545 = 654 – (5 × 2) = 644, 64 – (4 × 2) = 56 It is divisible by 7.

∴ 6545 is divisible by 7.

Rule for 13: Four times the last digit and adding it to the remaining number, �e result should be divisible by 13.

e.g., 169 = 16 + (9 × 4) = 16 + 36 = 52 It is divisible by 13

∴ 169 is divisible by 13

Rule for 17: Five times the last digit of the number and subtract it from the remaining number. �e result obtained should be either 0 or divisible by 17.

e.g., 2057 = 205 – 7 × 5 = 205 – 35 = 170 and 170 is divisible by 17.


Rule for 19: Double the last digit of given number and add it to the remaining number. �e result obtained should be divisible by 19.

e.g., 361 = 36 + (1 × 2) = 36 + 2 = 38 and 38 is divisible by 19.


Illustration 6: Find the smallest number of seven digits which is exactly divisible by 1111

Solution: Smallest number of seven digit = 1000000

On dividing by 1111 we get 100 as remainder

Number to be added = (1111 – 100) = 1011

Required number = 1000000 + 1011 = 1001011

1.10 Real Numbers

www.plancess.com

4.2 Divisibility of Algebraic Expressions�e expression an – bn is always divisible by (a – b) for all values of n (odd or even)

�e expression an – bn is exactly divisible by (a + b) when n is even.

�e expression an + bn is exactly divisible by (a + b) when n is odd.

�e expression an + bn never divisible by (a – b) whenever n is odd or even.

[where a b ].

PLANCESS CONCEPTS

NOTE:

1. �e product of n consecutive integers is divisible by n!

2. (a1 + a2 + ...+ an)! is divisible by a1! a2! ...... an!

Shivam Agarwal


Illustration 7: In a morning walk three persons step o� together, their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that they can cover the distance in complete steps?

Solution: Required minimum distance each should walk so that they can cover the distance in complete step is the L.C.M. of 80 cm, 85 cm and 90 cm

80 = 24 × 5; 85 = 5 x 17; 90 = 2 × 32 × 5

∴ LCM (80, 85, 90) = 24 × 32 × 51 × 171

LCM = 16 × 9 × 5 × 17 = 12240 cm = 122 m 40 cm.


www.plancess.com

SUMMARY

• Rational numbers and non-rational numbers taken together are called Real Numbers.

• Euclid’s division lemma or Euclid’s division algorithm

For any two positive integers a and b there exist unique integers q and r such that

a = bq + r, where 0 r < b.

• L.C.M. (a, b) a bH.C.F. (a, b)

=

• Divisibility Test of Numbers

A number is divisible by If

2 Unit digit should be 0 or even.

3 �e sum of digits of no. should be divisible by 3

4 �e no. formed by last 2 digits of given no. should be divisible by 4.

5 Unit digit should be 0 or 5.

6 No. should be divisible by 2 and 3 both.


9 Sum of digits of given no. should be divisible by 9

11 �e di�erence between sums of the digits at even and at odd places should be zero or multiple of 11.

25 Last 2 digits of the number should be 00, 25, 50 or 75.

• Divisibility of Algebraic Expressions:

(i) �e expression an – bn is always divisible by (a – b) for all values of n (odd or even)

(ii) �e expression an – bn is exactly divisible by (a + b) when n is even.

(iii) �e expression an + bn is exactly divisible by (a + b) when n is odd.

(iv) �e expression an + bn never divisible by (a – b) whenever n is odd or even.

(v) �e product of n consecutive integers is divisible by n!

(vi) (a1 + a2 + ...+ an)! is divisible by a1! a2! ...... an!

1.12 Real Numbers

www.plancess.com

SOLVED EXAMPLES

Example 1: Find 4 rational numbers between 1/5 and 1/6.

Solution: One rational number between 1/5 and 1/6 is 1 1 1 112 5 6 60

= 1 11 16 60 5

∴

Now, a rational number between 1/6 and 11/60 is 1 1 11 1 10 11 212 6 60 2 60 120

= =

1 21 11 16 120 60 5

Now, a rational number between 1160

and 15

is 1 11 1 1 11 12 232 60 5 2 60 120

= =

1 21 11 23 16 120 60 120 5

∴

Again one rational number between 23120

and 15

is 1 23 1 1 23 24 472 120 5 2 120 120

= =

1 21 11 23 47 16 120 60 120 240 5

∴

Hence, four rational number between 16

and 15

are 21 11 23 47, , ,

120 60 120 240.

Example 2: Find the L.C.M. of the polynomials 4 312(x x ) and 4 3 28(x 3x 2x ).

Solution: Let the given polynomials be denoted as p(x) and q(x) respectively

We have: 2 3p(x) 2 3 x (x 1)= ; 3 2q(x) 2 x (x 1) (x 2)=

Irreducible factors are 2, 3, x, x-1 and x-2.

�e respective highest exponents are 3, 1, 3, 1 and 1.

∴ L.C.M. = 3 3 32 3 x (x 1)(x 2) 24x (x 1)(x 2).=

Example 3: If (x+m) is the H.C.F of the functions 2x ax b and 2x cx d , then show that their L.C.M. is 3 2 2x (a c m)x (ac m ) x+m (a-m) (c-m)

Solution: Let 2x ax b (x m)(x p)=2x cx d (x m)(x q) where (x p) and (x q)

are prime to each other .=

2 2x ax b (x m)(x p) x x(m p) mp= =

2 2x cx d (x m)(x q) x x(m q) mq= = Equating coe�cient of x and constant terms in

the equations

∴ a = m + p; mp = b ...(i)

c = m + q; mq = d ...(ii)


www.plancess.com

From (i)p a m;From (ii)q c m;

=

=

�e L.C.M. of expression (i) and (ii) is (x+m)(x+p)(x+q)

=(x+m)(x+a-m)(x+c-m)

= 3 2x x [m a m c m] x[m(a m) m(c m) (a m)(c m)] m(a m)(c m)

3 2 2 2 2x x (a c m) x(am m cm m ac cm am m ) m(a m)(c m)=

3 2 2x (a c m)x (ac m )x m(a m)(c m).=

Example 4: �ree wheel can complete respectively 60, 36, 24 revolutions per minute. �ere is a red spot on each wheel that touches the ground at time zero. After how much time, all these spots will simultaneously touch again?

Solution: 1st wheel makes 1 revolutions per sec

2nd wheel makes 6/10 revolution per sec

3rd wheel makes 4/10 revolution per sec

In other words 1st, 2nd and 3rd wheel take 1, 5/3 and 5/2 seconds respectively to complete one revolution.

L.C.M of 1, 5/3 and 5/2= L.C.M of 1,5,55

H.C.M.of 1,3,2=

Hence, after every 5 seconds the red spots on all the three wheels touch the ground.

Example 5: When 2256 is divided by 17 then �nd the remainder.

Solution: When 2562 is divided by 17 then, 256 4 64

4 4

2 (2 )

2 1 (2 1)⇒ ⇒

By remainder theorem when f(x) is divided by (x+a), the remainder = f (-a)

Here f(x) = 4 64(2 ) and x= 42 and a=1

∴ Remainder =f (-1) = 64( 1) =1

Example 6: Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11 and 15 respectively.

Solution: Clearly, the required number is the H.C.F of the numbers

398 - 7=391,436 – 11= 425, and 542 – 15 =527.

First we �nd the H.C.F of 391 and 425 by Euclid’s algorithm as given below:

425=391 1+34

1.14 Real Numbers

www.plancess.com

391=34 11+17

34=17 2+0

Clearly, H.C.F of 391 and 425 is 17

Let us now the H.C.F of 17 and the third number 527 by Euclid’s algorithm:

527=17 31+0

�e H.C.F of 17 and 527 is 17. Hence, H.C.F of 391, 4250 and 527 is 17

Hence, the required number is 17.

Example 7: Two bills of Rs. 6075 and Rs. 8505 respectively are to be paid separately by cheques of same amount. Find the largest possible amount of each cheque.

Solution: Largest possible amount of cheque will be HCF (6075, 8505).

Applying Euclid’s division lemma to 8505 and 6075, we have 8505=6075 1+2430

Since remainder 0 again applying division lemma to 6075 and 2430

6075=2430×2+1215

Again remainder 0. So, again applying the division lemma to 2430 and 1215

2430=1215 × 2+0

Here, the remainder is zero

So, H.C.F=1215

�erefore, the largest possible amount of each cheque will be 1215.

Example 8: Prove that if x and y are odd positive integer, then 2 2x y is even but not divisible by 4.

Solution: We know that any odd positive integer is of the form 2q+1 for some integer q.

So, let x=2m+1 and y=2n+1 for some integers m and n.

⇒ 2 2 2 2x y (2m 1) (2n 1)=

2 2 2 2x y 4(m n ) 4(m n) 2⇒ =

2 2 2 2x y 4q 2,where q (m n ) (m n)⇒ = = 2 2x y⇒ is even and leaves remainder 2 when

divided by 4

2 2x y⇒ is even but not divisible by 4


www.plancess.com

Example 9: If 2 2a b is a prime number, show that 2 2a b a b= , where a, b are natural number.

Solution: 2 2a b (a b)(a b)=

2 2a b Is a prime number

One of the two factors =1

∴ a – b =1 … (1)

�e only divisor of a prime number are 1 and itself.

2 2a b = 1 (a+b) or 2 2a b = a+b

e.g., 2 23 2 = 5 (which is prime) 2 23 2 3 2 3,2 N⇒ =

1.16 Real Numbers

www.plancess.com

EXERCISE 1 – For School Examinations

Fill in the BlanksDirections: Complete the following statements with an appropriate word/term to be �lled in the blank space (s).

Q. 1. 3 2 5 is a / an ......... number number

Q. 2. An is a series of well-de�ned steps which gives a procedure for solving a type of problem.

Q. 3. A is a proven statement used for proving another statement.

Q. 4. L.C.M of 96 and 404 is .

Q. 5. H.C.F of 6, 72 and 120 is .

True / FalseDirections: Read the following statements and write your answer as true or false.

Q. 6. Given positive integers a and b, there exist whole numbers q and r satisfying a=bq +r, 0 ≤ r < b.

True False

Q. 7. Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

True False

Q. 8. If let x=p/q be a rational number, such that the prime factorization of q is of the form 2n 5m , where n, m are nonnegative integers. �en x has a decimal expansion which terminates.

True False

Q. 9. Any positive odd integer is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer.

True False

Q. 10. Cube of any positive integer is of the form 9m, 9m+1 or 9m+8.

True False

Q. 11. 1/0 is not rational.

True False

Q. 12. After rationalization the denominator of 5

3 2 2 3, we get the denominator as 7.

True False

Q. 13. �e number of irrational numbers between 15 and 18 is in�nite.

True False


www.plancess.com

Q. 14. is an irrational and 227

is a rational.

True False

Match the Following ColumnsDirections: Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in column I have to be matched with statements (p, q, r, s) in Column II

Q. 15. Match the following columns:

Column I Column II

(A) 16

is a (p) Non- terminating non-repeating decimal number

(B) 425

is a (q) Non-terminating repeating decimal number

(C) 3 6 is a (r) prime number

(D) 32 1331 (s) terminating decimal number

Very Short Answer QuestionsDirections: Give answer in one word or one sentence.

Q. 16. Find the H.C.F of 12576 and 4052 by using the fundamental theorem of Arithmetic.

Q. 17. Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.

Q. 18. �e length, breath and height of a room are 8 m 25 cm, 6 m 75 cm and 4m 50 cm respectively. Determine the longest rod which can measure the three dimensions of the room exactly.

Q. 19. A certain type of wooden board is sold only in lengths of multiples of 25 cm from 2 to 10 meters. A carpenters needs a large quantity of this type of boards in 1.65 meter length. For the minimum waste, �nd the lengths to be purchased.

Short Answer QuestionsDirections: Give answer in two to three sentences.

Q. 20. If the sum of two numbers is 1215 and their HCF is 81, �nd the number of such pairs.

Q. 21. Find H.C.F of 300, 540, and 890 by applying Euclid’s algorithm.

1.18 Real Numbers

www.plancess.com

Q. 22. A, B and C starts cycling around a circular path in the same direction at same time. Circumference of the path 1980m. If the speed of A is 330m/ min, speed of B is 198m/ min and C is 220 m/min and they start from the same point, then after what time interval they will be together at the starting point?

Q. 23. A certain number when divided by 899 leaves the remainder 63. Find the remainder when the same number is divided by 29.

Q. 24. A is the set of positive integers such that when divided by 2, 3, 4, 5 and 6 leaves the remainders 1, 2, 3, 4 and 5 respectively. How many integer (s) between 0 and 100 belongs to set A?

Q. 25. P is the product of all the prime numbers between 1 to 100. �en �nd the number of zeroes at the end of P.

Q. 26. Xn is either -1 or 1 and n 4; If 1 2 3 4 2 3 4 5 3 4 5 6 n 1 2 3x x x x x x x x x x x x ...... x x x x 0= then �nd the value of n.

Q. 27. �ere are two integers 34041 and 32506, when divided by a three-digit integer n, leave the same remainder. What is the value of n?

Q. 28. What is the remainder when 964 is divided by 6?

Q. 29. Four bells begin to roll together and toll respectively at intervals of 6, 5, 7, 10 and 12 seconds. How many times they will toll together in one hour excluding the one at the start?

Q. 30. H.C.F of 3240, 3600 and a third number is 36 and their L.C.M is 4 5 22 3 7 . Find the third number.

Q. 31. �e numbers 1 to 29 are written side by side follows 1236547891011………..2829. If the number is divided by 9, then what is the remainder?

Long Answer QuestionsDirections: Give answer in four to �ve sentences.

Q. 32. A sweet seller has 420 kaju bar�s and 130 badam bar�s. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the maximum number of bar� that can be placed in each stack for this purpose?

Q. 33. Find the H.C.F of 81 and 237 and express it as a linear combination of 81 and 237.

Q. 34. Prove that 2 5 is irrational.

Q. 35. Find the unit’s digit in the product 35 71 557 3 11 .


www.plancess.com

EXERCISE 2 – For Competitive Examinations

Multiple Choice QuestionsDirections: �is section contains multiple choice questions. Each question has 4 choices (a), (b), (c) and (d) out of which ONLY ONE is correct.

Q. 1. When 2562 is divided by 17 the remainder would be

(a) 1 (b) 16 (c) 14 (d) None of these

Q. 2. �e sum of three non-zero prime numbers is 100. One of them exceeds the other by 36. �en the number is

(a) 73 (b) 91

(c) 67 (d) 57

Q. 3. H.C.F. of 3 2(x 3x 2) and (x 4 x 3) is

(a) (x-1) (b) 2(x 1)

(c) ( ( 2x 1) x ) (d) (x 1)(x 3)

Q. 4. If the two numbers when divided by a certain divisor give remainder 35 and 30 respectively and when their sum is divided by the same divisor, the remainder is 20, then the divisor is

(a) 40 (b) 45

(c) 50 (d) 55

Q. 5. In order that the number 1 y 3y 6 be divisible by 11, the digit y should be

(a) 1 (b) 2

(c) 5 (d) 6

Q. 6. Find the least number which when divisible by 12, leaves a remainder of 7, when divided by 15, leaves a remainder of 10 and when divided by 16, leaves a remainder of 11

(a) 115 (b) 235

(c) 247 (d) 475

Q. 7. If n 4n( 1) ( 1) 0= , then n is

(a) Any positive integer (b) Any negative integer

(c) Any odd natural number (d) Any even natural number

Q. 8. �e number 13 103 3 is divisible by

(a) 2 and 3 (b) 3 and 10

(c) 2, 3 and 10 (d) 2, 3 and 13

1.20 Real Numbers

www.plancess.com

Q. 9. A number lies between 300 and 400. If the number is added to the number formed by reversing the digits, the sum is 888 and if the unit’s digit change places, the new number exceeds the original number by 9. �en the number is

(a) 339 (b) 341

(c) 378 (d) 345

Q. 10. Number that has to be added to 345670 in order to make it divisible by 6 is

(a) 2 (b)4

(c) 5 (d) 6

Q. 11. �e greatest number which can divide 1854, 1866 and 2066 leaving the same remainder 2 in each case is

(a) 4 (b) 6

(c) 12 (d) None

More than One CorrectDirections: �is section contains multiple choice questions. Each question has 4 choices (a), (b), (c) and (d) out of which ONLY ONE OR MORE may be correct.

Q. 12. Which of the following is always false?

(a) �e sum of two distinct irrational numbers is irrational

(b) �e rationalizing factor of a number is unique

(c) Every irrational number is a surd

(d) Any surd of the form n na b can be rationalized by a surd of the form n na b , where n aand n b are surds.

Q. 13. �e possible numbers that can be formed by using unit digit and ten’s digit in (274 243 131)lies between

(a) 23 and 45 (b) 23 and 44

(c) 26 and 46 (d) 25 and 50

Q. 14. �e product of unit digit in 95 58(7 3 ) and 95 58(7 3 ) is

(a) cube of 2 (b) lies between 6 and 10

(c) 6 (d) lies between 3 and 6

Passage Based QuestionsDirections: Study the given passage(s) and answer the following questions.

Q. 15. LCM of several fractions = LCM of their numeratorsHCFof their denominators


www.plancess.com

HCF of several fractions = HCFof their numeratorsLCM of their denominators

i. �e L.C.M. of the fractions 5 15 25, , and

16 24 8 is

(a) 548

(b) 58

(c) 7548

(d) 758

ii. �e H.C.F of 2 6 8, , and

5 25 35is

(a) 25

(b) 245

(c) 2175

(d) 24175

iii. �e H.C.F. of the fractions 821

, 1235

and 327

is

(a) 4105

(b) 1927

(c) 47

(d) 5109

Assertion and Reason Directions: Each of these questions contains an Assertion followed by Reason. Read them carefully and answer the questions on the basis of following options. You have to select the one that best describes the two statements.

Q. 16. Assertion: �e H.C.F. of two numbers is 16 and their product is 3072. �en their L.C.M=162.

Reason: If a, b are two positive integers, then H.C.F L.C.M. = a b.

Q. 17. Assertion: 2n n is divisible by 2 for every positive integer.

Reason: 2 is not a rational number.

Q. 18. Assertion: 2n n is divisible by 2 for every positive integer n.

Reason: If x and y are odd positive integers, from 2 2x y is divisible by 4.

Multiple Matching QuestionsDirections: Following question has four statements (A, B, C and D) given in Column I and four statements (p, q, r and s) in Column II. Any given statement in Column I can have correct matching with one or more statement(s) given in Column II. Match the entries in Column I with entries in Column II.

1.22 Real Numbers

www.plancess.com

Q. 19. Match the columns:

Column I Column II

(A) 3 6 9

551

2 5 7

(p) a prime number

(B) Product of ( 5 3) and ( 5 3) (q) is an irrational number

(C) 5 4 (r) is a terminating decimal representation

(D) 3 4

422

2 5

(s) a rational number

(t) is a non-terminating but repeating decimal representation

(u) is non-terminating and non-recurring decimal representation

Subjective QuestionsDirections: Answer the following questions.

Q. 20. Show that there is no positive integer n for which n 1 n 1 is rational.

Q. 21. If n is positive integer, let S(n) denote the sum of the positive divisors of n, including n, G(n) is

the greatest divisor of n. If H(n) = G(n)S(n)

then, which of the following is the largest H(2010) or H(2011)?


www.plancess.com

SOLUTIONS

Exercise 1– For School Examinations

Fill in the Blanks1. Rational 2. Algorithm 3. Lemma 4. 9696 5. 6

True / False6. True 7. True 8. True 9. True 10. True

11. True 12. False 13. True 14. True

Match the Following Columns15. (A) (q) (B) (s) (C) (p) (D) (r)

Very Short Answer Questions16. H.C.F. 22 = 4

17. 240 = 24×3×5, 1024=210

18. 75 cm.

19. 1.65m =165 cm, Required length =L.C.M of 25 and 165=825 cm =8.25 m

Short Answer Questions20. (1134, 81), (1053, 162), (891, 324), (648, 567)

21. H.C.F of (300, 540, 890) =10

22. �ey will meet after 30 min.

23. Divided =Divisor Quotient +Remainder

Let the quotient=1, then Divided=899+63=962

Hence, remainder =5 when same no. is divided by 29.

24. Only one integer between 0 and 100 belong to A i.e. 59.

25. �ere are only 2 prime numbers 5 and 2 between 1 and 100 which when multiplied will give zero in the end. �us, there will be only one at the end of the product of given number.

26. Every term in the question is either 1 or -1. In order to have zero the number of terms must be even. Note that there are n number of terms. (Since the �rst term in each product varies from

1 nx to x ). So n has to be even.

1.24 Real Numbers

www.plancess.com

27. Let the common remainder be x. �en numbers (34041 - x) and (32506 - x) would be completely divisible by n. Hence the di�erence of the numbers (34041-x) and or (34041-x) will also be divisible by n or (34041-x-32506+x)=1535 will also be divisible by n. Now, using options we �nd that 1535 is divisible by 307.

28. From this we know that remainder for any power of 4 will be 4 only.

29. L.C.M of 6, 5, 7, 10 and 12=420 seconds = 4207

60= minutes. �erefore, in one hour

(60 minutes,) they will fall together 8 times 607

exclusive the one at the start.

30. H.C.F. 2 22 3= ; 4 5 2L.C.M 2 3 7=

3 4 4 2 21st number 2 3 5; 2nd number 2 3 5= = Observing the above situation, we conclude that the third number must be

2 2 3 2 2 5 2x 2 3 3 7 2 3 7= =

31. Hence, the given number remains 3 as remainder when divided by 9.

Long Answer Questions32. Now, let us use Euclid’s algorithm to �nd their H.C.F. We have:

420 130 3 30;130 30 4 10;30 10 3 0

===

So, the H.C.F of 420 and 130 is 10.

�erefore, the sweet seller can make stacks of 10 for both kinds of bar�.

33. Given integers are 81 and 237 such 81<237. Applying division lemma to 81 and 237,

we get 237=81 2+75 … (i)

Since, the remainder 75 0. So, consider the divisor 81 and the remainder 75 and apply division lemma to get 81=75 1+6 …(ii)

We consider the new divisor 75 and the new reminder 6 and apply division lemma to get

75=6 12+3 …(iii)

We consider the new divisor 6 and the new reminder 3 and apply division lemma to get

6=3 2+0 …(iv)

�e remainder at this stage is zero. So, the divisor at this stage or the remainder at the earlier stage i.e. , 3 is the H.C.F of 81 and 237.

To represent the H.C.F as a linear combination of the given two numbers, we start from the last but one step and successively eliminate the previous remainders as follows: From (iii), we have

3=75-(6 12)


www.plancess.com

⇒ 3=75-(81-75) 12

[Substituting 6=81-75 1 obtained from (ii)]

3 75 12 81 12 753 13 75 12 81

[Substituting 75 237 81 2 obtained from(i)]3 13 (237 81 2) 12 813 13 237 26 81 12 813 13 237 38 813 237x 81y, where x 13 and y 38.

⇒ =⇒ =

=⇒ =⇒ =⇒ =⇒ = = =

34. Let us assume on the contrary that 2 5 is rational number. �en, there exist co-prime positive integers a and b such that

22

2 2

2 2

a a2 5 2 5

b b

a2 ( 5) [Squaring both sides]

b

a 3b2 2 is a rational number

2ab

a 3ba, b are integers is rational

2ab

= ⇒ =

⇒ =

⇒ = ⇒

�is contradicts the fact that 2 is rational. So, our assumption is wrong.

Hence, 2 5 is a irrational.

35. Units digits in 4(7 ) =1. �erefore, units digits in 4 8(7 ) i.e., 327 will be 1. Hence, units digits in 35(7) 1 7 7 7 3= = Again, units digit in the expansion of

4 17 68(3 ) (3) 1= = ⇒ Units digits in the expansion of 71(3 ) 1 3 3 3 7= =

And units digits in the expansion of 55(11) 1= Hence, units digits in the expansion of35 71 557 3 11 3 7 1 1= =

1.26 Real Numbers

www.plancess.com

Exercise 2 – For Competitive Examinations

Multiple Choice Questions1. (a) When 2562 is divided by 17 then,

256 4 64

4 4

2 (2 )

2 1 (2 1)⇒ ⇒

By remainder theorem when f(x) is divided by x+a the remainder =f(-a)

Here 4 64 4f(a) (2 ) and x 2 and a 1= = = ∴Remainder =f(-1)= 64( 1) 1=

2. (c) Let �rst prime number is 2 and second is x, then third number will be (x+36) .

�ree prime numbers are 2, 31, and 67

Largest number is 67.

3. (a) 3 2(x 3x 2) (x 1)(x 1)(x 2) and (x 4 x 3) (x 1)(x 3)= =

4. (b)Divisor = 1 2 3r r r 35 30 20 45= =

5. (c) Use test of 11 after putting y=5

6. (b)

7. (c) 4n 2 2n( 1) [( 1) ] 1= =

8. (d) 13 10 10 3 10 103 3 3 (3 1) 3 (26) 2 13 3= = =

13 10Hence, 3 3 is divisible by 2, 3, and 13

9. (d) Sum is 888 ⇒ unit’s digits should add up to 8. �is is possible only for D option as “3”+”5”=”8”.

10. (a) On dividing the number 345670 by 6, we get 4 as the remainder.

So, 2 must be added to the given number.

11. (a)

More than One Correct12. (a, b, c, d)

13. (a, b)

In the product of (274 243 131) Unit digit will be 2 and Ten’s digit will be 4. We can make only two possible numbers from 2 and 4.

So the numbers are 24 or 42.

14. (a, b)

Unit digit in 95(7 ) = Unit digit in 4 23[(7 ) 7]

= Unit digit in 37 (as unit digit in 47 =1) = Unit digit in 343

Unit digit in 583 = Unit digit in 4 4 2(3 ) 3 = Unit digit is 9 [as unit digit 43 1= ]


www.plancess.com

So unit digit in 95 58(7 3 ) = Unit digit in (343 - 9) = Unit digit in 343 = 4

Unit digit ( 95 587 3 ) = Unit digit in (343 + 9) = Unit digit in 352 = 2

So the product is 4 2 8=

Passage Based Questions

15. i. (d) L.C.M. of 5 15 25, , and

16 24 8= LCM of their numerators

HCFof their denominators

L.C.M of, 15 and 25 is 75. H.C.F of 16, 24 and 8 is 8.

�e H.C.F of the given fractions= 758

ii. (c) H.C.F of the fractions = HCFof their numeratorsLCM of their denominators

H.C.F. of 2, 6 and 8 is 2. L.C.M. of 5, 25 35 is 175.

�us, the H.C.F. of the given fractions = 2175

iii. (a) H.C.F of the fractions is = H.C.F. of 8,12,32 4L.C.M.of 21,35,7 105

=

Assertion and Reason16. (d) Here reason is true [standard result]

Assertion is false. 3072192 162

16= ∴ (d) holds

17. (b)

18. (a)

Multiple Matching Question19. (A) (t,s);(B) (p,s);(C) (q,u); (D) (r,s)

Subjective Questions20. We assume that there is a positive integer n for which n 1 n 1 is rational, and equal to a

bwhere a and b are positive integers.

So, n 1 n 1 = ab

…..(1) b 1,

a n 1 n 1⇒ =

Rationalizing RHS by multiplying Nr and Dr by n 1 n 1 , we get

1.28 Real Numbers

www.plancess.com

n 1 n 1ba 2= or n 1 n 1b

a 2= n 1 n 1⇒ = 2b

a …(2)

Adding equation (1) and (2), we get 2 2a 2b

n 12ab

⇒ =

…(3)

Subtracting (2) from (1), we get 2 2a 2b

n 12ab

⇒ =

…(4)

From (3) and (4), we have n 1 and n 1 are both rational as a and b are both positive integer.

�is is possible only when both n+1 and n-1 are perfect square of some positive integer n.

(n-1) and (n+1) di�er by 2. But two perfect squares never di�er by 2, hence (n-1)and (n+1) both cannot be perfect square. So, here is no positive number n for which n 1 + n 1 is a rational number.

21. Let the positive divisors of n be 1, a, b, c… and n itself. Let S(n) =sum of positive divisors of n, including n. =1+a+b+c+……..+n.

And g(n) = greatest divisor of n=n (itself )

Given: H(n) = G(n) nS(n) 1 a b c ....... n

=

Now, For H(n)to be largest, 1+a+b+c…..+n should be minimum. In this series 1 and n can never be removed so if we ignore a, b, c, …. , (a+b +c+….)=0. Hence, we get the minimum value of S(n)as 1+n.

Hence, H(n) is maximum when n is a prime number because it has only two positive divisor 1 and n.

Now we are given two values, 2010 and 2011, of which 2011 is prime. Hence, H(2011) is the largest.

Documents

1. REAL NUMBERS - TopperLearning