IIT JEE 2019 Leaked Question

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Thanks to Martin Gardner

1) Prove that a^n - b^n is divisible by (a-b)
( you can juggle your way through this one without too much difficulty. For n being even, you split it as you know how so you only have to prove it for the case of n being odd. https://math.stackexchange.com/questions/188657/why-is-an-bn-divisible-by-a-b)

2) Prove that 10^(2k+1) + 1  (i.e., 11, 1001, 100001, etc) is divisible by 11 and that 10^2k - 1 (i.e., 99, 9999, etc) is divisible by 11

3) Prove that a palindromic prime number must have an odd number of digits.

Thanks of course to Martin Gardner :)