Page 62 - Start Up Mathematics_6

Example 29: In which of the following expressions, has prime factorization been done?

(a) 36 = 2 × 3 × 6 (b) 40 = 2 × 2 × 2 × 5
Solution: (a) 36 = 2 × 3 × 6
 6 is not a prime number. ∴ prime factorization is not done.
(b) 40 = 2 × 2 × 2 × 5
 all the factors are prime numbers. ∴ prime factorization is done.
Example 30: Determine if 25,155 is divisible by 45.
Solution: To determine if 25,155 is divisible by 45, we must test divisibility by 5 and 9.
25,155 is divisible by 5 as the digit at its ones place is 5. It is divisible by 9 as the
sum of the digits of 25,155 is 2 + 5 + 1 + 5 + 5 = 18, which is divisible by 9.
Since, 25,155 is divisible by both 5 and 9 therefore, it is divisible by its product 45.
Example 31: 45 is divisible by both 3 and 5. It is also divisible by 3 × 5 = 15. Similarly, a number
is divisible by both 6 and 9. Can we say that the number must also be divisible by
6 × 9 = 54? If not, give an example to justify your answer.
Solution: No, we cannot say that the number will be divisible by 54 if it is divisible by 6 and 9.
Let the number which is divisible by both 6 and 9 be 90. However, 90 is not divisible
by 54.
Example 32: I am the smallest number having five different prime factors. Can you find me?
Solution: To get the smallest number having five different prime factors, we need to find the
product of five smallest prime numbers.
Required prime numbers are 2, 3, 5, 7 and 11.
Therefore, the smallest number is 2 × 3 × 5 × 7 × 11 = 2,310.

EXERCISE 3.5

1. Express the following in terms of prime factorization:
(a) 2,904 (b) 1,500 (c) 3,969 (d) 9,317 (e) 2,907 (f) 4,641

2. Write the greatest 3-digit number and express it in terms of its prime factors.
3. Write the smallest 5-digit number and express it in terms of its prime factors.
4. Find prime factors of 20,570.

Highest Common Factor (HCF)

The highest common factor (HCF) of two or more given numbers is the highest (or greatest) of
their common factors. In other words, HCF is the greatest number which exactly divides two or
more given numbers.

The HCF of numbers is also known as GCD (greatest common divisor).

HCF by listing method

Example 33: Find the HCF of 12 and 18.
Solution: All possible factors of 12 are 1 , 2 , 3 , 4, 6 and 12.

All possible factors of 18 are 1 , 2 , 3 , 6 , 9 and 18.
The common factors of the given numbers are 1, 2, 3 and 6.
The HCF of 12 and 18 is the highest of their common factors, i.e., 6.

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