For every positive integer n, prove that 7ⁿ – 3ⁿ is divisible by 4.

Let P(n) = 7ⁿ – 3ⁿ

For n = 1,

P (1) = 7¹ - 3¹

= 4

Which is divisible by 4.

Thus P (1) is true.

Consider P(K) to be true for some natural number k.

i.e. P(K) = 7^k – 3^k is divisible by 4.

We now have to prove P(K+1) is divi9sible by 4 whenever P(K) is true.

So,

7^K+1 – 3^k+1 = 7^K+1 – 7.3^k + 7.3^k+ 3^k+1

= 7(7^k – 3^k) + (7-3)3^k

= 7(4d) + (7-3)3^k

= 7(4d) + 4.3^k

= 4(7d + 3^k)

Hence, we can see that 7^K+1 – 3^k+1 is divisible by 4.

Thus P(K+1) is true when P(K) is true.

∴ By mathematical induction the given statement is true for all n.