Difference between revisions of "2024 AMC 10A Problems/Problem 15"
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We wish to maximize both <math>P</math> and <math>Q,</math> so we maximize <math>Q+P</math> and minimize <math>Q-P.</math> It follows that | We wish to maximize both <math>P</math> and <math>Q,</math> so we maximize <math>Q+P</math> and minimize <math>Q-P.</math> It follows that | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
− | Q+P=1280, \ | + | Q+P&=1280, \ |
− | Q-P=2, | + | Q-P&=2, |
\end{align*}</cmath> | \end{align*}</cmath> | ||
from which <math>(P,Q)=(639,641).</math> | from which <math>(P,Q)=(639,641).</math> | ||
+ | |||
+ | Note that <math>M=P^2-1213=Q^2=3773\equiv1-3\equiv8\pmod{10},</math> so the units digit of <math>M</math> is <math>\boxed{\textbf{(E) }8}.</math> | ||
~MRENTHUSIASM | ~MRENTHUSIASM |
Revision as of 16:06, 8 November 2024
Problem
Let be the greatest integer such that both and are perfect squares. What is the units digit of ?
Solution
Let and for some positive integers and We subtract the first equation from the second, then apply the difference of squares: Note that and have the same parity, and
We wish to maximize both and so we maximize and minimize It follows that from which
Note that so the units digit of is
~MRENTHUSIASM
See also
2024 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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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