Difference between revisions of "2024 AMC 10A Problems/Problem 19"
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| + | ==Problem== | ||
| + | The first three terms of a geometric sequence are the integers <math>a,\,720,</math> and <math>b,</math> where <math>a<720<b.</math> What is the sum of the digits of the least possible value of <math>b?</math> | ||
| + | <math>\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21</math> | ||
| + | ==Solution== | ||
| + | For a geometric sequence, we have <math>ab=720^2=2^8 3^4 5^2</math>, and we can test values for <math>b</math>. We find that <math>b=768</math> and <math>a=675</math> works, and we can test multiples of <math>5</math> in between the two values. Finding that none of the multiples of 5 divide <math>720^2</math> besides <math>720</math> itself, we know that the answer is <math>7+6+8=\boxed{\textbf{(E)}21</math>. | ||
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| + | ~eevee9406 | ||
Revision as of 15:52, 8 November 2024
Problem
The first three terms of a geometric sequence are the integers 
and 
where 
What is the sum of the digits of the least possible value of 
Solution
For a geometric sequence, we have 
, and we can test values for 
. We find that 
and 
works, and we can test multiples of 
in between the two values. Finding that none of the multiples of 5 divide 
besides 
itself, we know that the answer is $7+6+8=\boxed{\textbf{(E)}21$ (Error compiling LaTeX. Unknown error_msg).
~eevee9406
