Arithmetic LCM and HCF

8/12/2019 Arithmetic LCM and HCF

1/90


2/90

Pratyush @ ToughestQuestions

Number SystemsNatural & Whole numbers

Integers & FractionsRational & Irrational numbers

Prime & Composite numbers


3/90


Divisibility Rules

A number is divisible by 2 if

That number is an evennumber (the last digit of the

number is an even number)


4/90


5/90


Divisibility Rules


The last two digits of that

number are divisible by 4


6/90


Divisibility Rules


The last digit is 0 or 5.


7/90


Divisibility Rules


That number is divisible byboth 3 and 2 (an even number,

whose digits add up to anumber divisible by 3)


8/90


Divisibility Rules


The last three digits of that

number are divisible by 8


9/90


Divisibility Rules


The sum of all the digits of the

number is divisible by 9.


10/90


Divisibility Rules


The difference of the Sum of

alternate digits to be 0 ordivisible by 11


11/90


Addition and Multiplication rulesfor Even and Odd numbersEven + Even = Even

Even * Even = EvenEven + Odd = Odd

Even * Odd = EvenOdd + Odd = Even

Odd * Odd = Odd


12/90


Can Even / Odd be aninteger?

Depends on the numbers.

For example, 28/7 is an integer.


13/90


Can Odd / Odd be aneven integer?

Never


14/90


Divide an even number with

an odd number. If remainderexists, is it odd or even?

18/5 - Remainder is 3 - odd

30/ 7 - Remainder is 2 - evenNothing can be said.


15/90


Indices


16/90


OddOdd= ?

Odd


17/90


EvenOdd= ?

Even


18/90


a3* a2= ?

a5

Rule: am* an= a m+n


19/90


(a3)2= ?

a6

Rule: (am) n= a m n


20/90


a3/a2= ?

aRule: am/ an= a m - n


21/90


(ab)3= ?

a3 b3

Rule: (ab)m= a m bm


22/90


a- 3 = ?

1/ a3

Rule: (a) - m= 1 / a m


23/90


What if a is a fraction or

a negative number?

The rules are still applicable


24/90


LCM / HCF

Wh t th lti l f


25/90

Pratyush @ Toughest Questions

What are the multiples of

6 and 9?6: 6, 12, 18, 24, 30

9: 9, 18, 27, 36

Common Multiples: 18, 36, ..

LCM: 18


26/90


What are the factors of 6

and 9?6: 1, 2, 3, 6

9: 1, 3, 9

Common Factors: 1, 3

HCF: 3


27/90


Find the LCM and HCF of2835 & 3456.

LCM = 362880, HCF = 27


28/90


Find the LCM and HCF of

3/4 and 1/2.

LCM = 3/2, HCF = 1/4


29/90


The LCM of two fractions

a/b and c/d is

(LCM of a & c) / (HCF of b & d)


30/90


The HCF of two fractions

a/b and c/d is

(HCF of a & c) / (LCM of b & d)


31/90


Exercise


32/90


What is the fraction form

of 0.33333....?

1/3


33/90



of 0.44444.. ?

4/9


34/90



of 0.12121212.. ?

12/99


35/90


What is the last digit of71*72*73*...*79?

The product of 72 and 75 willend in a zero.

So, the last digit of the productwill be zero.

d b th di it f


36/90


a and b are the digits of a

two digit number. Then(ab - ba) is always a

multiple of ...ab = 10a + b, ba = 10b + a

ab - ba = 9a - 9b

It will always be a multiple of 9

f f


37/90


A sum of five successive

whole numbers is alwaysdivisible by ...

Let the numbers be x, x+1, x+2,

x+3, x+4. Their sum is 5x + 10.It will always be divisible by 5.

When divided by 8 any (odd


38/90


When divided by 8, any (oddnumber)2gives ___ as a

remainder

Square of odd = (2a+1)2= 4a 2+ 4a + 1 = 4 (a 2+a) + 1

(a 2+a) will be even, for all a.

4(a 2+a) will be multiple of 8

Th d t


39/90


The product

67

x 353

x 1110

has howmany prime factors?

6 has 2 prime factors - 2 & 3So 67has 14 prime factors

Similarly 353has 6 and

1110has 10 prime factors.


40/90


Mentally calculate the

square of 65

Take the tens digit, 6, add 1to it.

Multiply the two 6*7 = 42Append 25 to this - 4225.


41/90


Mentally calculate 58*52Take the tens digit, 5, add 1 to

it.

Multiply the two 6*5 = 30Append 8*2 = 16 to this - 3016.


42/90


What is 245 x 25?

= (250*25) - (5*25)

= 6250 - 125= 6125


43/90


What is 204?

142= 196

Calculate (14.1)2, (14.2)2andso on (algebraically).

So the square root is between14 and 15, but closer to 14.


44/90

Pratyush @ Toughest

Questions

What is 66% of 999 ?

= approx. 66% of 1000

= 660


45/90

Pratyush @ Toughest

Questions

Which is larger 103*97 or

107*93?

103*97 = 1002

- 32

107*93 = 1002- 72

So 103*97 is bigger


46/90

Pratyush @ Toughest

Questions

Which is greater - 16/66

or 25%?

16/66 < 16/64So, the fraction is less than 1/4,

or less than 25%So 25%is bigger


47/90

Pratyush @ Toughest

Questions

What is 107*107?

= (100 +7)2

= 1002+ 2*100*7 + 72

= 10,000 + 1400 + 49 = 11449


48/90


49/90

Pratyush @ Toughest

Questions

Which is bigger 999- 998

or 998?

999- 998= 998(9-1)

= 8*998

So 999- 998is bigger.


50/90

Pratyush @ Toughest

Questions

If I first your marks by

10% and then them by10%, which are more:your earlier marks or

changed marks?The earlier marks are more.


51/90

Pratyush @ Toughest

Questions

What is the remainder

when 1012+ 5 is dividedby 3?

The sum of digits of 1012+5 is 6.

Since this is divisible by 3,

the remainder is 0.

I 5353 3333 di i ibl b


52/90


Is 5353- 3333divisible by

10?Last digits of a number ending

with 3 will be..31= 3 32= 9

33= 7 34= 1

.


53/90


After every 4thpower the series

will repeat.

So 353and 333will both end in

3.

And the difference will be

divisible by 10.

3 b d i l


54/90


3 boys go round a circular

field in 3, 4 and 5 minutesrespectively. They start from

the same place at the sametime. When will they be at

the same place again?They will meet after 60 minutes


55/90

Pratyush @ Toughest

Questions

Revision

C i f i


56/90

Pratyush @ Toughest

Questions

Conversion of recurring

decimal 0.abcabcabc .. tofraction is

abc / 999


57/90

Pratyush @ Toughest

Questions

Shortcut for square of a5?

a(a+1),25


58/90

Pratyush @ Toughest

Questions

LCM * HCF =

Product of the two numbers


59/90

Pratyush @ Toughest

Questions

The LCM of two fractions

a/b and c/d is

(LCM of a & c) / (HCF of b & d)


60/90

Pratyush @ Toughest

Questions

The HCF of two fractions

a/b and c/d is

(HCF of a & c) / (LCM of b & d)


61/90

Pratyush @ Toughest

Questions

Averages

A t i f h


62/90

Pratyush @ Toughest

Questions

A countrys airforce has

18 squadrons and a totalof 558 aircraft. What is the

average number of aircraftin a squadron?

Average = 558 / 18 = 31


63/90

Pratyush @ Toughest

Questions

Average

Total of class________________________

number of elements in class

If th l i


64/90

Pratyush @ Toughest

Questions

If the annual p.c. income

in X is $130 and that in Yis $90, what is the

average p.c. income of Xand Y taken together?

= 130 + 90 / 2 = 220 / 2 = 110


65/90

Pratyush @ Toughest

Questions

If it is given that Ys

population is half of Xs?= (130* 2x + 90 * x) / 3x =116.7

If ith 4 t d


66/90

Pratyush @ Toughest

Questions

If a car with 4 tyres and a

spare traveled 20000 km,and all tyres have been

used equally, what wasthe mileage of each tyre?

80000 / 5 = 16000

Which is the smallest


67/90

Pratyush @ Toughest

Questions

Which is the smallest

whole number, whenmultiplied with 576, makes

a perfect cube?576 = (23) * (23) * (32)

Multiply by 3 to get a perfect

cube

What should be minimum


68/90

Pratyush @ Toughest

Questions

What should be minimum

number of boys so thatthey can be arranged in

the group of 20, 25, 35 or40 ?

The LCM of 20, 25, 35 and 40

= 1400.

A batsman's avg for


69/90

Pratyush @ Toughest

Questions

A batsman s avg. for

some innings was 42.5.Next 4 innings he scored

45, 48, 41 & 56, and hisavg. for all the innings

became 43.75. How manyinnings in all did he play ?

Total till start = 42 5x


70/90

Pratyush @ Toughest

Questions

Total till start = 42.5x.New = 42.5x + (45+48+41+56).Avg. = (42.5x + 190)/ (x+4) =43.75

Solve for x, answer = 12.

The average age of Ajit


71/90

Pratyush @ Toughest

Questions

The average age of Ajit

and his wife when theymarried 5 years ago was

24. What is the presentaverage age of the family

if they have a 2 year-oldson?


72/90

Pratyush @ Toughest

Questions

The couples current age =

2(24 + 5) = 58.Including son, total = 60.

Average = 60/3 = 20.

Jill earned $ 1300 last


73/90

Pratyush @ Toughest

Questions

Jill earned $ 1300 last

month. If her basic salaryis $ 1000 more than her

overtime, how much didshe earn in overtime?

Let overtime be x;x + (1000+x) = 1300. x = 150.

A number is created by


74/90

Pratyush @ Toughest

Questions

A number is created by

writing the first 54numbers:

12345678............. 5354.This, when divided by 8,

will leave a remainder of?Remainder will be 354-352 = 2

What is the number if


75/90

Pratyush @ Toughest

Questions

What is the number, if

increased by 7, issimultaneously divisible

by 27, 21, 35 and 77?

LCM of above numbers =21*99*5 = 10395

number = 10395 7 = 10388


76/90


77/90

Mr X enters a positive


78/90

Pratyush @ Toughest

Questions

Mr. X enters a positive

number y in a calculatorand then goes on

pressing the square root() key repeatedly. What

will happen?


79/90

Pratyush @ Toughest

Questions

For any positive number, the

larger the n, the closer thevalue of nthroot to 1.

The product of all


80/90

Pratyush @ Toughest

Questions

The product of all

integers from 1 to 100 willhave -- number of zeroes

at the end.One each for each multiple of 5from 5 to 100; four more for 25,50, 75, and 100. Answer = 24.

Three wheels rotate at 60


81/90

Pratyush @ Toughest

Questions

Three wheels rotate at 60,

36 and 24 rpm. Each hasa red spot on its rim,

which coincides at thestart. The red spots will be

at this position again afterhow many seconds?


82/90

Pratyush @ Toughest

Questions

The wheels make 1, 0.6 and

0.4 rev/sec respectively.LCM of the three is 6 sec.

Th LCM f t b


83/90

Pratyush @ Toughest

Questions

The LCM of two numbers

is 12 times their HCF. Thesum of the HCF and LCM

is 403. If one of thenumbers is 93, find theother.


84/90

Pratyush @ Toughest

Questions

Let HCF = x. LCM = 12x.

LCM+HCF =13x = 403.

So x = 31

LCM*HCF = Product

Other number = 12*31*31 / 93= 124

A, B & C sit together for a


85/90

Pratyush @ Toughest

Questions

A, B & C sit together for a

meal. A has 3 loaves, Bhas 4 and C has none.

They share the loavesequally and C pays $21.

How much should B get?


86/90

Pratyush @ Toughest

Questions

Loaves eaten/person = 7/3,A contributed 3 - 7/3, = 2/3,

and B contributed 4 - 7/3 = 5/3.So, B gets 5/7 * 21 = $15

X engaged Y at a salary


87/90

Pratyush @ Toughest

Questions

X engaged Y at a salaryof $ 2400 for 30 days. Yforfeits $ 100/day that he

didnt work. At the end,neither owed the other

any money. How manydays did Y work?


88/90

Pratyush @ Toughest

Questions

80x = 100(30-x).

So x = 3000/180=16 and 2/3days

A two digit number "ab"


89/90

Pratyush @ Toughest

Questions

A two digit number abis squared. The square ofthis number is of the form

"ccb". If we know that"ccb" is greater than 300,

then what is the value ofb?


90/90

172= 289. 352> 1000.

So, the number is between 17and 35.

"b" can take only values0,1,5,6.

The only number that fits is 21.

Documents

Arithmetic LCM and HCF