Difference between revisions of "2020 AMC 10A Problems/Problem 24"

Problem

Let $n$ be the least positive integer greater than $1000$ for which$$\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.$$What is the sum of the digits of $n$?

$\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 21\qquad\textbf{(E) } 24$

Solution

Because we know that $gcd(63, n+120)=21$, we can write $n+120\equiv0(mod 21)$. Simplifying, we get $n\equiv6(mod 21)$. Similarly, we can write $n+63\equiv0(mod60)$, or $n\equiv-3(mod60)$. Solving these two modular congruences, $n\equiv237(mod 420)$ which we know is the only solution by CRT (Chinese Remainder Theorem). Now, since the problem is asking for the least positive integer greater than $1000$, we find the least solution is $n=1077$. However, we are have not considered cases where $gcd(63, n+120) =63 or$gcd(n+63, 120) =120$.$1077+120\equiv0(mod63)$so we try$n=1077+420=1497$.$1497+63\equiv0(120) so again we add another $420$ to $n$. It turns out that $n=1497+420=1917$ does indeed satisfy the conditions, so our answer is $1+9+1+7=\boxed{\textsf{(C) } 18}$.

Video Solution

https://youtu.be/tk3yOGG2K-s - $Phineas1500$