Difference between revisions of "2025 AMC 8 Problems/Problem 6"
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The sum of the residues of these numbers modulo <math>4</math> is <math>-1+0+1+2+3=5 \equiv 1 \pmod 4</math>. Hence, the number being subtracted must be congruent to <math>1</math> modulo <math>4</math>. The only such answer is <math>\boxed{\textbf{(C)}~17}</math>. | The sum of the residues of these numbers modulo <math>4</math> is <math>-1+0+1+2+3=5 \equiv 1 \pmod 4</math>. Hence, the number being subtracted must be congruent to <math>1</math> modulo <math>4</math>. The only such answer is <math>\boxed{\textbf{(C)}~17}</math>. | ||
~cxsmi | ~cxsmi | ||
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== Solution 4 == | == Solution 4 == | ||
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== Solution 5== | == Solution 5== | ||
| − | We can use the fact that <math>(a + b) \mod n = [(a \mod n) + (b \mod n)] \mod n</math>. Notice that 15, 16, 17, 18, 19<math> dividing by 4 have remainders < | + | We can use the fact that <math>(a + b) \mod n = [(a \mod n) + (b \mod n)] \mod n</math>. Notice that <math>15, 16, 17, 18, 19</math> dividing by 4 have remainders <math>3, 0, 1, 2, 3</math>. Their sum is 9. It is easy to see that <math>9-1=8</math> is divisible by 4 and so C (17) is the correct answer. |
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== Video Solution 1 by Cool Math Problems == | == Video Solution 1 by Cool Math Problems == | ||
Latest revision as of 19:15, 17 September 2025
Contents
Problem
Sekou writes the numbers 
After he erases one of his numbers, the sum of the remaining four numbers is a multiple of 
Which number did he erase?
Solution 1
The sum of all five numbers is 
. Since is 
more than a multiple of 
, the number being subtracted must be more than a multiple of . Thus, the answer is 
.
~Gavin_Deng
Solution 2
The sum of the residues of these numbers modulo is 
. Hence, the number being subtracted must be congruent to modulo . The only such answer is .
~cxsmi
Solution 4
Since the sum of 
is odd, we immediately exclude B and D. We further note that if A is true, then E is true. Hence, the answer is C.
Solution 5
We can use the fact that ![$(a + b) \mod n = [(a \mod n) + (b \mod n)] \mod n$](http://latex.artofproblemsolving.com/9/9/1/9912596152140f8dc08c47fe3f58d8cce38dcb10.png)
. Notice that 
dividing by 4 have remainders 
. Their sum is 9. It is easy to see that 
is divisible by 4 and so C (17) is the correct answer.
Video Solution 1 by Cool Math Problems
https://youtu.be/BRnILzqVwHk?si=1KwyuFBUDqdMtC6t&t=2
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=jTTcscvcQmI
Video Solution 3
Video Solution 4 by Thinking Feet
Video Solution 5 by Daily Dose of Math
Video Solution(Quick, fast, easy!)
~MC
See Also
| 2025 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
