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

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== Solution ==
 
== Solution ==
  
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We know that <math>gcd(63, n+120)=21</math>, so we can write <math>n+120\equiv0(mod 21)</math>. Simplifying, we get <math>n\equiv6(mod 21)</math>. Similarly, we can write <math>n+63\equiv0(mod60)</math>, or <math>n\equiv-3(mod60)</math>. Solving these two modular congruences, <math>n\equiv237(mod 420)</math> 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 <math>1000</math>, we find the least solution is <math>n=1077</math>. However, we are have not considered cases where <math>gcd(63, n+120) =63</math> or <math>gcd(n+63, 120) =120</math>. <math>1077+120\equiv0(mod 63)</math> so we try <math>n=1077+420=1497</math>.  
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We know that <math>gcd(63, n+120)=21</math>, so we can write <math>n+120\equiv0(\mod 21)</math>. Simplifying, we get <math>n\equiv6(\mod 21)</math>. Similarly, we can write <math>n+63\equiv0(\mod 60)</math>, or <math>n\equiv-3(\mod 60)</math>. Solving these two modular congruences, <math>n\equiv237(\mod 420)</math> 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 <math>1000</math>, we find the least solution is <math>n=1077</math>. However, we are have not considered cases where <math>gcd(63, n+120) =63</math> or <math>gcd(n+63, 120) =120</math>. <math>{1077+120}\equiv0(mod 63)</math> so we try <math>n=1077+420=1497</math>. <math>{1497+63}\equiv0(\mod 120)</math> so again we add <math>420</math> to <math>n</math>. It turns out that <math>n=1497+420=1917</math> does indeed satisfy the original conditions, so our answer is <math>1+9+1+7=\boxed{\textsf{(C) } 18}</math>.
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<math>1497+63\equiv0(mod 120)</math> so again we add <math>420</math> to <math>n</math>. It turns out that <math>n=1497+420=1917</math> does indeed satisfy the conditions, so our answer is <math>1+9+1+7=\boxed{\textsf{(C) } 18}</math>.
 
  
 
== Video Solution ==
 
== Video Solution ==

Revision as of 01:27, 1 February 2020

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

We know that $gcd(63, n+120)=21$, so we can write $n+120\equiv0(\mod 21)$. Simplifying, we get $n\equiv6(\mod 21)$. Similarly, we can write $n+63\equiv0(\mod 60)$, or $n\equiv-3(\mod 60)$. 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 $gcd(n+63, 120) =120$. ${1077+120}\equiv0(mod 63)$ so we try $n=1077+420=1497$. ${1497+63}\equiv0(\mod 120)$ so again we add $420$ to $n$. It turns out that $n=1497+420=1917$ does indeed satisfy the original conditions, so our answer is $1+9+1+7=\boxed{\textsf{(C) } 18}$.

Video Solution

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

See Also

2020 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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