Question 12
The number of ordered pairs $$(m,n)$$ of integers such that $$1\le m,n\le100$$ and $$m^n\cdot n^m$$ leaves a remainder of $$1$$ when divided by $$4$$ is
Solution
Both numbers must be odd. For odd exponents, the product is congruent to $$mn$$ modulo $$4$$, so the remainder is $$1$$ exactly when both numbers are congruent to $$1$$ modulo $$4$$ or both are congruent to $$3$$ modulo $$4$$. There are $$25$$ numbers of each type from $$1$$ to $$100$$. Hence the number of ordered pairs is $$25^2+25^2=1250$$.
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