Maths Olympiad Prep

Track / Stage 5 / 350 of 400 #950 of 1964

Problem 950

AIME late
Number theory Difficulty 5.7 Prove it

On his birthday, Dadya Fyodor's parents give him an amount of money equal to the product of his father's age and his mother's age. Could it have happened that the amounts received by him in 2010 and 2011 ended with the same digit, and the amount received in 2012 was divisible by 10?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

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Official solution

Answer: It could. Solution. For example, if in 2010 Uncle Fyodor's father was 38 years old, and his mother was 31 years old. Note. All possible examples are when the ages of Uncle Fyodor's parents in 2010 end in the digits 3 and 6, or when they end in 1 and 8. There are no other examples.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.