Difference between revisions of "2022 AMC 12A Problems/Problem 16"
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Therefore, the answer is <math>4+1+6+1+6=\boxed{\textbf{(D) 18}}</math>. | Therefore, the answer is <math>4+1+6+1+6=\boxed{\textbf{(D) 18}}</math>. | ||
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+ | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
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+ | ==Video Solution== | ||
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+ | https://youtu.be/ZmSg0JYEoTw | ||
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) |
Revision as of 21:47, 11 November 2022
Problem
A \emph{triangular number} is a positive integer that can be expressed in the form , for some positive integer
. The three smallest triangular numbers that are also perfect squares are
,
, and
. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?
Solution
We have .
If
is a perfect square, then it can be written as
,
where
is a positive integer.
Thus, .
Because and
are relatively prime, the solution must be in the form of
and
, or
and
, where in both forms,
and
are relatively prime and
is odd.
The four smallest feasible in either of these forms are
.
Therefore, .
Therefore, the answer is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)