JMO -- DIvisibility

Which are the two numbers less than 260 that exactly divides

 2³² – 1?

JMO – VII and VIII

Answer:

2³² = {2⁸}⁴ = 256⁴

Now 256⁴ – 1 can be written as 256⁴ – 1⁴

This is in (aⁿ – xⁿ) format, which is divisible by (a + x) and (a – x) if n is even.

So, 256⁴ – 1⁴ is divisible by (256 + 1) and (256 – 1), i.e., 257 and 255.

Since both are less than 260, it satisfies the condition in the question. Hence answer.