Question

5^2n+1 is divisible by 24 for all n belongs to N. Prove using PMI.​

Answer 1

Answer:

As you have asked for a method other than Induction method,

I will prove this problem using the concept of congruences.

Since x^k - 1 always has x - 1 as a factor,

x^k - 1 = (x - 1)(x^k-1 + x^k-2 + · · · + x + 1)

the expression (5^2n) - 1 = (5^2)^n - 1 always has 5^2- 1 = 24 as a factor.

That’s your proof.

Step-by-step explanation:

We can use the binomial expansion of (24+1)n to prove the required.

52n−1=25n−1

25n−1=(24+1)n−1

=1+ terms with 24 as a factor −1

= divisible by 24