Maths Olympiad Prep

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Number theory Difficulty 5.2 AIME, harder Find the answer

13. Find all positive integer pairs (a,b)(a, b) that satisfy (a,b)=10,[a,b]=100(a, b)=10,[a, b]=100.

A number or a short expression. Spacing and $ signs are ignored.

Solution

13. (a/10,b/10)=1,[a/10,b/10]=10.a/10,b/10(a / 10, b / 10)=1,[a / 10, b / 10]=10 . a / 10, b / 10 can only take the values 1,2,5,101,2,5,10, and satisfy the above conditions. Therefore, we get the following table:
\begin{tabular}{|c|cccc|}
\hlinea/10a / 10 & 1 & 10 & 2 & 5 \\
\hlineb/10b / 10 & 10 & 1 & 5 & 2 \\
\hline
\end{tabular}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.