Question 2

The sum of the 3-digit numbers $$35a$$ and $$4b7$$ is divisible by 36. Find all possible pairs $$(a, b)$$.


Correct Answer: (5 3)(1 7)

Solution

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Writing the sum as $$300+50+a+400+10b+7 = 757+10b+a$$, this must be a multiple of 36. Testing nearby multiples of 36 shows $$10b+a=35$$ or $$10b+a=71$$ are the only values fitting single-digit a and b. These give the pairs $$(a,b)=(5,3)$$ and $$(a,b)=(1,7)$$.

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