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Found problems: 2

2023 AMC 10, 23
Tags: digit problems
Positive integer divisors $a$ and $b$ of $n$ are called [i]complementary[/i] if $ab=n$. Given that $N$ has a pair of complementary divisors that differ by $20$ and a pair of complementary divisors that differ by $23$, find the sum of the digits of $N$. $\textbf{(A) } 11 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$

2024 AIME, 14
Tags: digit problems , number theory
Let $b\ge 2$ be an integer. Call a positive integer $n$ $b$-[i]eautiful[/i] if it has exactly two digits when expressed in base $b$ and these two digits sum to $\sqrt{n}$. For example, $81$ is $13$-[i]eautiful[/i] because $81 = \underline{6} \ \underline{3}_{13} $ and $6 + 3 = \sqrt{81}$. Find the least integer $b\ge 2$ for which there are more than ten $b$-[i]eautiful[/i] integers.