MathLabs

Problem 2

For a positive integer kk, call an integer a pure kk-th power if it is mkm^k for some integer mm. Show that for every positive integer nn there exist nn distinct positive integers whose sum is a pure 20092009-th power and whose product is a pure 20102010-th power.
Step 7 of 7: Verify the product
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∏ai=(n! sn(k2−1)/(k+1))k+1\prod a_i=(n!\,s^{n(k^2-1)/(k+1)})^{k+1}
Detailed analysis

Because k+1k+1 divides k2−1k^2-1, ∏ai=sn(k2−1)(n!)k+1=(n!sn(k2−1)/(k+1))k+1\prod a_i=s^{n(k^2-1)}(n!)^{k+1}=(n!s^{n(k^2-1)/(k+1)})^{k+1}, a pure 20102010-th power.