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Number Theory
A prime number is a natural number greater than 1 that has exactly two factors (or divisors), itself and 1.
Prime numbers less than 50{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47}
* “2” is the only EVEN Number
285The number ends in 0 or 5.
5
844 since 44 4
The number formed by the last two digits of the number is divisible by 4.
4
846 since 8 + 4 + 6 = 18
The sum of the digits of the number is divisible by 3.
3
846The number is even.2
ExampleTestDivisible by
730The number ends in 0.10
846 since 8 + 4 + 6 = 18
The sum of the digits of the number is divisible by 9.
9
3848since 848 8
The number formed by the last three digits of the number is divisible by 8.
8
846The number is divisible by both 2 and 3.
6
ExampleTestDivisible by
Write the prime factorization of 663.
The final quotient 17, is a prime number, so we stop. The prime factorization of 663 is 3 •13 •17
13
3
17
221
663
The greatest common divisor of two or more integers can be obtained in three steps:
STEP 1: Find the prime factorization of each integer. (Use Division Method)
375 = 3 × 53 525 = 3 × 52 × 7
STEP 2: List the common prime divisors (factors) with the least power of all the given integers.
375 = 3 × 53 = 3 × 52 × 5 525 = 3 × 52 × 7 = 3 × 52 × 7
Common Prime Divisors (Factors) with Least Power: 3 and 52
STEP 3: Multiply the common prime divisors (factors) to find the greatest common divisor (factor).
3 × 52 = 75
GCD (GCF) of 375 and 525 = 75
The least common multiple (denominator) of two or more integers can be obtained in three steps:
STEP 1: Find the prime factorization of each integer. (Use Division Method)
4 = 22 10 = 2 × 5 45 = 32 × 5
STEP 2: List the prime divisors (factors) with the greatest power of all the given integers.
4 = 22 10 = 2 × 5 45 = 32 × 5
Prime Divisors (Factors) with Greatest Power: 22, 32, and 5
STEP 3: Multiply the prime divisors (factors) to find the least common multiple (denominator).
22 × 32 × 5 = 180
LCM of 4, 10 and 45 = 180
Find the GCD of 63 and 105. 63 = 32 • 7 105 = 3 • 5 • 7
Smallest exponent of each factor:3 and 7
So, the GCD is 3 • 7 = 21.
Find the LCM of 63 and 105. 63 = 32 • 7105 = 3 • 5 • 7
Greatest exponent of each factor:32, 5 and 7
So, the LCM is 32 • 5 • 7 = 315.
Find the GCD and LCM of 48 and 54. Prime factorizations of each:
48 = 2 • 2 • 2 • 2 • 3 = 24 • 354 = 2 • 3 • 3 • 3 = 2 • 33
GCD = 2 • 3 = 6
LCM = 24 • 33 = 432
Evaluate:a) (3)(4) b) (7)(5)
c) 8 • 7 d) (5)(8)
Solution:a) (3)(4) = 12 b) (7)(5) = 35
c) 8 • 7 = 56 d) (5)(8) = 40
Evaluate:a) b)
c) d)
Solution:a) b)
c) d)
72
9
72
9
72
8
72
8
72
98
72
98
72
89
72
89
Fractions are numbers such as:
The numerator is the number above the fraction line.
The denominator is the number below the fraction line.
1
3,
2
9, and
9
53.
Convert to an improper fraction.
5
7
10
(10 5 7)
10
50 7
10
57
10
5
7
10
Convert to a mixed number.
The mixed number is
7 236
21
33
26
21
5
236
7
33
5
7.
Evaluate the following.
a)
b)
2
3
7
16
2
3
7
16
27316
14
48
7
24
1
3
4
2
1
2
13
4
2
1
2
7
45
2
35
84
3
8
Evaluate the following.
a)
b)
2
3
6
7
2
3
6
7
2
37
6
2736
14
18
7
9
5
8
4
5
5
8
4
5
5
85
4
5584
25
32
Add:
Subtract:
4
9
3
9
4
9
3
9
4 3
9
7
9
11
16
3
16
11
16
3
16
11 3
16
8
16
1
2
Evaluate:
Solution:
7
12
1
10.
7
12
1
10
7
125
5
1
106
6
35
60
6
60
29
60
The principal (or positive) square root of a number n, written is the positive number that when multiplied by itself, gives n.
For example,
16 = 4 since 44 =16
49 = 7 since 77 = 49
n
Simplify:a)
b)
ab a b, a 0, b 0
40 410 4 10 2 10 2 10
125 255 25 5 5 5 5 5
40
125
Simplify: Simplify: 4 7 3 7
4 7 3 7
(4 3) 7
7 7
8 5 125
8 5 125
8 5 25 5
8 5 5 5
(8 5) 5
3 5
Simplify:
6 54
6 54 654 324 18
Move the decimal point in the original number to the right or left until you obtain a number greater than or equal to 1 and less than 10.
Write each number in scientific notation.a) 1,265,000,000.
1.265 109 Decimal Point to left is
b) 0.0000000004324.32 1010 Decimal Point to right is
Write each number in decimal notation.a) 4.67 105
467,000
b) 1.45 10–7
0.000000145
Large Number move to left and is a number
Small Number move to right and is a number
Addition a + b = b + a for any real numbers a and b.
Multiplication a • b = b • a for any real numbers a and b.
8 + 12 = 12 + 8 is a true statement. 5 9 = 9 5 is a true statement.
Note: The commutative property does not hold true for subtraction or division.
Addition (a + b) + c = a + (b
+ c),
for any real numbers a, b, and c.
Multiplication (a • b) • c = a • (b • c),
for any real numbers a, b, and c.
(3 + 5) + 6 = 3 + (5 + 6) is true.
(4 6) 2 = 4 (6 2) is true.
Note: The associative property does not hold true for subtraction or division.
Distributive property of multiplication over additiona • (b + c) = a • b + a • cfor any real numbers a, b, and c.
Example: 6 • (r + 12) = 6 • r + 6 • 12 = 6r + 72
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