Some number theory training questions: RMO and INMO

Question 1:

Let us write an arbitrary natural number (for example, 2583), and then add the squares of its digits. (2^{2}+5^{2}+8^{2}+3^{2}=102). Next, we do the same thing to the number obtained. Namely, 1^{2}+0^{2}+2^{2}=5. Now proceed further in the same way:

5^{2}=25, 2^{2}+5^{2}=29, 2^{2}+9^{2}=85, \ldots.

Prove that unless this procedure leads to number 1 (in which case, the number 1 will, of course, recur indefinitely), it must lead to the number 145, and the following cycle will repeat again and again:

145, 42, 20, 4, 16, 37, 58, 89.

Question 2:

Prove that the number 5^{5k+1} + 4^{5k+2} + 3^{5k} is divisible by 11 for every natural k.

Question 3:

The number 3^{105} + 4^{105} is divisible by 13, 49, 181 and 379, and is not divisible by either 5 or by 11. How can this result be confirmed?


Nalin Pithwa.

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