Page 60 - Start Up Mathematics_6

EXERCISE 3.4

1. Find the common factors of:

(a) 10 and 15 (b) 16 and 20 (c) 21 and 35 (d) 3, 6 and 9
(e) 8, 12 and 20 (f) 45, 60 and 105
2. Find the first three common multiples of:

(a) 8 and 12 (b) 6 and 9 (c) 10 and 25
3. Write all the numbers less than 90 which are common multiples of 5 and 6.
4. A number is divisible by 15. By what other numbers will that number be divisible?

5. Which of the following pairs of numbers are co-prime?
(a) 13 and 16 (b) 16 and 21 (c) 25 and 105
6. A number is divisible by 5 and 8 both. By which other number will that number be always
divisible?

Some Properties of Divisibility
As discussed earlier, we generalize some very useful properties of divisibility which help us to
test the divisibility of numbers.

1. If a number ‘A’ is divisible by number ‘B’, then number ‘A’ is also divisible by each factor
of ‘B’.
Example: 24 is divisible by 6. Clearly, 24 is also divisible by each factor of 6, i.e., 1, 2
and 3.
2. If a number is divisible by two co-prime numbers, then it is also divisible by their product.

Example: 120 is divisible by two co-prime numbers 5 and 6. Clearly, 120 is also divisible by
the product of 5 and 6, i.e., 30.
3. If two given numbers ‘A’ and ‘B’ are divisible by a number ‘C’, then their sum ‘A + B’
is also divisible by ‘C’.

Example: Numbers 15 and 35 are divisible by 5. Clearly, 15 + 35 = 50 is also divisible by 5.
4. If two given numbers ‘A’ and ‘B’ are divisible by a number ‘C’, then their difference
‘A – B’ is also divisible by ‘C’.
Example: Numbers 35 and 15 are divisible by 5. Clearly, 35 – 15 = 20 is also divisible by 5.

Prime Factorization

Consider the composite number 40. We can write 40 = 5 × 8. As 8 is Do you know?
again composite we can write 8 as 2 × 2 × 2. Hence, 40 = 2 × 2 × 2 The fundamental theorem
× 5. Alternatively, we can write 40 = 4 × 10. Now 4 can be written of Arithmetic states that
as 2 × 2 and 10 as 2 × 5, hence, 40 = 2 × 2 × 2 × 5. In both the cases, every composite number
we obtain the same factorization 2 × 2 × 2 × 5. In this factorization, can be factorized into prime
the factors 2 and 5 are prime numbers. This process of expressing a factors in a unique way.
number as a product of prime factors is called prime factorization.

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