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Prove that among any $n$ integers, one can choose several whose sum is divisible by $n$. Consider the partial sums $S_k = a_1 + \dots + a_k$. Look at the remainders modulo $n$. If any remainder is 0, we are done. If not, by the Pigeonhole Principle, two sums $S_i$ and $S_j$ ($i < j$) must have the same remainder. Their difference $S_j - S_i$ is divisible by $n$.

Systematic mastery of integration by substitution, integration by parts, and the use of partial fractions. pure maths lee peng yee pdf link

A solid O-Level pure math background is essential for Junior College math. Conclusion Prove that among any $n$ integers, one can