Introduction to Modern Algebra Number Theory PROBLEM Prove

[Introduction to Modern Algebra & Number Theory]

PROBLEM: Prove that for any integer a, 3 divides a(a+4)(a-1).

Solution

We will prove it by mathematical induction

For a=1, a(a+4)(a-1) = 0, 3 divides 0

So for a=1, it is true.

Le the relation holds for a=n,

Then, 3 divides n(n+4)(n-1) = n(n² +3n -4)

So we need to prove it for a=n+1,

Replacing n by n+1

(n+1)(n+5)n = n(n² + 6n + 5)

= n(n² + 3n + 3n -4 + 9)

=  n(n² + 3n -4 + 3n + 9)

n² + 3n -4 is divided by 3 as proved above and 3n+9 is also clearly divided by 3

So n² + 3n -4 + 3n + 9 = n² + 6n + 5 is divided by 3

Hence by mathematical induction for any integer a, 3 divides a(a+4)(a-1).