2013 AMC 10B Problem 25

Let the last two base-5 digits of $N$ be $d_1 d_0$ and the last two base-6 digits be $e_1 e_0$. Then $N \equiv 5d_1 + d_0 \pmod{25}$ and $N \equiv 6e_1 + e_0 \pmod{36}$, while LeRoy's sum satisfies $S \equiv 10(d_1 + e_1) + (d_0 + e_0) \pmod{100}$.

Units digit. $N \equiv d_0 \pmod 5$ gives $2N \equiv 2d_0 \pmod{10}$. We need $d_0 + e_0 \equiv 2d_0 \pmod{10}$, so $e_0 = d_0$: the remainders of $N$ mod $5$ and mod $6$ agree. Hence $N = 30m + r$ with $0 \le r \le 4$. Since $d_0 \le 4$, the units sum $2d_0$ produces no carry.

Tens digit. Write $N = 25k + 5d_1 + d_0$, so $2N = 50k + 10d_1 + 2d_0$. The condition becomes $10(d_1 + e_1) \equiv 50k + 10d_1 \pmod{100}$, i.e. $e_1 \equiv 5k \pmod{10}$. As $e_1 \le 5$: $e_1 = 0$ if $k$ is even and $e_1 = 5$ if $k$ is odd.

Now $e_1$ is the base-6 tens digit, $\lfloor N/6 \rfloor \bmod 6 = 5m \bmod 6$, which is $0$ when $m \equiv 0$ and $5$ when $m \equiv 1 \pmod 6$. And $k = \lfloor N/25 \rfloor$ depends only on $m$ since $r \le 4$.

For $100 \le N \le 999$, $m$ runs from $4$ to $33$. Check $m \equiv 0$ (need $k$ even): $m = 6, 12, 18, 24, 30$ give $k = 7, 14, 21, 28, 36$, so $m = 12, 24, 30$ work. Check $m \equiv 1$ (need $k$ odd): $m = 7, 13, 19, 25, 31$ give $k = 8, 15, 22, 30, 37$, so $m = 13, 31$ work.

Five values of $m$, each with five choices of $r$: $25$.

The answer is $\boxed{\textbf{(E)}\ 25}$.

Only the last two digits of each representation influence the last two digits of $S$, so everything reduces to congruences modulo $25$, $36$ and $100$. Attacking the units digit before the tens digit is the key organizing move: the first condition collapses the search to $N \equiv r \pmod{30}$, and the second is a parity check on $\lfloor N/25 \rfloor$.

• Treating the base-5 and base-6 strings as base-10 numbers without separating the units-digit condition from the tens-digit condition, or forgetting carries are impossible here.
• Testing only a few sample values of $N$ and extrapolating; the valid $N$ come in blocks of five ($360$-$364$, $390$-$394$, $720$-$724$, $900$-$904$, $930$-$934$).