Prime Factorization

Playing With Numbers of Class 6

Prime factorization

prime factorization

In each case we get prime number as factors at last. It is called prime factorization.

When A number is expressed as the product of prime numbers, then it is known as its prime factorization.

Illustration 3:

Write the prime factors of following:

(i) 36 (ii) 52 (iii) 99 (iv) 100

Sol.

(i) 36 = 2 × 2 × 3 × 3

(ii) 52 = 2 × 2 × 13

(iii) 99 = 3 × 3 × 11

(iv) 100 = 2 × 2 × 5 × 5

CO–Prime

Two numbers having only 1 as a common factor are called co–prime.

Illustration 4:

Are 25 and 21 co-prime ?

Sol.

21 = 1 × 3 × 7

25 = 1 × 5 × 5

Common factor – 1

Hence 21 and 25 are co-prime.

Test for divisibility of numbers

1. Divisibility by 2: The numbers having 0, 2, 4, 6 and 8 at its ones place are divisible by 2.

e.g., 10, 12, 100, 248 …… are divisible by 2.

2. Divisibility by 3: A number is divisible by 3 if sum of its digits is a multiple of 3.

e.g., 39, 63, 135, are divisible by 3.

3. Divisibility by 4: A number with 2 or more digits is divisible by 4 if the number formed by its last two digits (i.e., ones and tens) is divisible by 4.

e.g., 96, 108, 224 ….. are divisible by 4.

4. Divisibility by 5: A number which has either 0 or 5 in its ones place is divisible by 5.

e.g., 105, 255, 350, 455, 675 …… are divisible by 5.

5. Divisibility by 6: A number which is divisible by 2 and 3 both then it is divisible by 6 also.

e.g., 24, 48, 96, 108 ……. are divisible by 6.

6. Divisibility by 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

e.g., 512, 9216, 1024, 7648……. are divisible by 8.

7. Divisibility by 9: A number is divisible by 9 if the sum of the digits of the number is divisible by 9.

e.g., 81, 927, 1089 ……. are divisible by 9.

8. Divisibility by 10: If a number has 0 in the ones place then number is divisible by 10.

e.g., 100, 210, 250 ……… are divisible by 10.

9. Divisibility by 11: If the difference of the sum of its digits at odd places and sum of the digits at even places is equal to zero or a multiple of 11, then that number is divisible by 11.

e.g., 2431 sum of odd place digits = 1 + 4 = 5

Sum of even place digits = 2 + 3 = 5

Difference of the sum of its digits at odd places and sum of the digits at even places = 5 – 5 = 0

Since difference is zero hence 2431 is divisible by 11.

Some more divisibility rules:

(i) If a number is divisible by another number then it is divisible by each of the factors of that number.

(ii) If a number is divisible by two co-prime numbers then it is divisible by their product also.

(iii) If two numbers are divisible by a number then their sum and difference is also divisible by that number.

Highest common factor

The highest common factor (HCF) of two or more given number is the highest (or greatest) of their common factors. It is also known as greatest common divisor (GCD).

Illustration 5:

(i) Find HCF of 14, 20 and 50.

Sol.

14 = 2 × 7

20 = 2 × 2 × 5

50 = 2 × 5 × 5

HCF = is 2

(ii) Find greatest common divisor of 18, 24 and 36.

Sol.

18 = 2 × 3 × 3

24 = 2 × 3 × 2 × 2

36 = 2 × 3 × 2 × 3

HCF = 2 × 3 = 6

Lowest common multiple (LCM)

The lowest common multiple (LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiple.

Illustration 6:

(i) Find LCM of 12 and 24.

Sol.

12 = 2 × 2 × 3

24 = 2 × 2 × 3 × 2

LCM = 2 × 2 × 3 × 2 = 24

2 12, 24
2 6, 12
2 3, 6
3 3, 3
  1, 1

LCM = 2 × 2 × 3 × 2 = 24

(ii) Find lowest common multiple of 24, 30 and 36

Sol.

24 = 2 × 2 × 3 × 2

30 = 2 × 3 × 5

36 = 2 × 2 × 3 × 3

LCM = 2 × 2 × 2 × 3 × 3 × 5

= 8 × 9 × 5 = 360.

2 24, 30, 36
2 12, 15, 18
2 6, 15, 9
3 3, 15, 9
3 1, 5, 3
5 1, 5, 1

LCM = 2 × 2 × 2 × 3 × 3 × 5 = 360

H.C.F of (a, b) × L.C. M of (a, b) = a × b

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