Question

If a is divisible by neither 2 nor 3, show that a2-1 is divisible by 24.

Answer 1

Answer:

a can be express as 3k or 3k+1 or 3k+2

Since a is not divisible by 2 and 3 so a can express as 3k+2 only

Then a^2–1=(3k+2)^2-1=3(3k^2+4k+1) which is divisible by 3

Again a can be of the form 8k or 8k+1 or… or 8k+7,but as a is not divisible by 2 and 3 so a is of the form 8k+3 or 8k+5 or 8k+7

As a=8k+3, a^2-1=8(8k^2+6k+1) which is divisible by 8

As a=8k+5,a^2-1=8(8k^2+10k+3) which is divisible by 8

As a=8k+7,a^2-1=8(8k^2+14k+6) which is divisible by 8

So (a^2-1) is divisible by both 3 & 8 and gcd(3,8)=1

So 24 divide (a^2-1)