Page 45 - Maths Skill - 6

Prime Time 43

PROPERTIES OF DIVISIBILITY
Property 1
Let a, b, c be three numbers. If a is divisible by b and b is divisible by c, then a is divisible by c. We define the
property with an example as:
If 108 is divisible by 12 and 12 is divisible by 6, then 108 is divisible by 6.

Property 2

If a and b are the pair of co-prime numbers, a is a factor of c and b is also a factor of c, then a × b must be a factor
of c.

For example; let us take two co-prime numbers 4 and 7. 4 is a factor of 84 and 7 is also a factor of 84. Therefore,
4 × 7 = 28 is also a factor of 84.

Property 3
If a is a factor of two numbers b and c, then a must be a factor of (b + c).
For example; 6 is a factor of 36 and 72. Then 6 is also a factor of 36 + 72 = 108.

Property 4
If a is a factor of two numbers b and c, then a must be a factor of their difference, i.e., b – c (b > c) or c – b (c > b).

For example; 7 is a factor of 28 and 14, then 7 is a factor of 28 – 14 (14 < 28).
For example; 7 is a factor of 28 and 42, then 7 is a factor of 42 – 28 (42 > 28).

TO DETERMINE IF A GIVEN NUMBER IS PRIME OR NOT
Suppose the given number is 197, and we want to test whether the number is prime or not.

We know 15 × 15 = 225 and 225 is greater than 197.
We divide 197 by prime numbers less than 15, i.e., 2, 3, 5, 7, 11 and 13 by using divisibility tests.
If 197 is not divisible by any one of the prime numbers less than 15, then it is a prime number; otherwise it is not
a prime number.

Important Results

Finding Prime Numbers between 100 and 200
If the given number is not divisible by any one of the prime numbers less than 15, then it is prime; otherwise it is
not a prime number.

Finding Prime Numbers between 100 and 400

If the given number is not divisible by any one of the prime numbers less than 20, then it is prime; otherwise it is
not a prime number.

Let’s Attempt

Example 1: In each of the following numbers, replace the sign ‘*’ by the smallest digit to make them divisible
by the given number:
(i) 308* by 2 (ii) 267*89 by 3 (iii) 45*8* by 5
(iv) 56315* by 8 (v) 9375*2 by 9 (vi) 4259*48 by 11