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Putnam 1963 Let $\mathbb N$ be the set of positive integers, and let $f:\mathbb N\to\mathbb N$ be a strictly increasing function such that $f(2)=2$ and $f(m)f(n)=f(mn)$ for all positive integers $m,n$ such that $\gcd(m,n)=1$. Find all such functions $f$. Watered down version: Let $\mathbb N$ be the set of positive integers, and let $f:\mathbb N\to\... 2 For what natural$n$does there exist a square composed of$n$squares? Example: 1, 4, and 6 are valid, but one cannot construct a square from 2, 3, or 5 squares. Proof: 3 Here is another one. Prove that the power of$13$can be writen as a sum of two squares. I will give two profs of it. First one is more involved and includes lemma $$(a^2+b^2)(x^2+y^2)= (ax+by)^2+(bx-ay)^2$$ yet second takes step 2 and it is much more elegant. 1.st proof Base:$n=1$, then$13 = 2^2+3^2$and we are done. We know that$13^n = a^2+b^2$... 2 A simple consequnce of: Postage Stamp Problem, which states that for any two relatively prime positive integers$m,n$, the greatest integer that cannot be written in the form$am + bn$for nonnegative integers$a, b$is$mn-m-n$, is that every natural number greater or equal to$mn-m-n+1$can be writen in a form$am+bn$for some$a,b\$. And for some ...