Permutation and Combination
Permutation and Combination
Allen Star Batch
Grade 11
Question:
The number of ordered pairs $(m, n), m, n \in \{1,2,....50\}$ such that $6^m + 9^n$ is a multiple of 5:
Step-by-Step Solution
Key Concept: Using complex roots of unity (specifically $i$) as substitutions separates alternating sums of coefficients, with periodicity based on $n \bmod 4$.
Substituting $x = i$ into $i^n = (a_0 - a_2 + a_4 - ...) + i(a_1 - a_3 + a_5 - ...)$ gives $a_0 - a_2 + a_4 - ... + i(a_1 - a_3 + a_5 - ...)$. The result depends on $n \bmod 4$: if $n$ is even the result is 0, if $n = 4k+1$ then $a_0 - a_2 + a_4 - ... = 1$, and if $n = 4k+3$ then $a_0 - a_2 + a_4 - ... = -1$.
Correct Answer: 2