# Prime Factorization

## Playing With Numbers of Class 6

**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.

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**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

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**CO–Prime**

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

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**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.

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**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.

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**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.

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**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).

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**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

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**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.

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**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**