Difference between revisions of "2025 AMC 8 Problems/Problem 16"
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==Solution== | ==Solution== | ||
| − | Note that for no two numbers to differ by <math>10</math>, every number chosen must have a different units digit. To make computations easier, we can choose <math>(1, 2, 3, 4, 5)</math> from the first group and <math>(16, 17, 18, 19, 20)</math> from the second group. Then the sum evaluates to <math>1+2+3+4+5+16+17+18+19+20 = \boxed{\text{(C)\ 105}}</math>. | + | Note that '''for no two numbers to differ by <math>10</math>, every number chosen must have a different units digit'''. To make computations easier, we can choose <math>(1, 2, 3, 4, 5)</math> from the first group and <math>(16, 17, 18, 19, 20)</math> from the second group. Then the sum evaluates to <math>1+2+3+4+5+16+17+18+19+20 = \boxed{\text{(C)\ 105}}</math>. |
~Soupboy0 | ~Soupboy0 | ||
| − | + | ||
| + | ~ Edited by [[User:Aoum|Aoum]] | ||
== Another Way To Compute == | == Another Way To Compute == | ||
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~Bepin999 | ~Bepin999 | ||
| − | + | ==Video Solution by Pi Academy== | |
| − | = Video Solution by Pi Academy = | ||
https://youtu.be/Iv_a3Rz725w?si=E0SI_h1XT8msWgkK | https://youtu.be/Iv_a3Rz725w?si=E0SI_h1XT8msWgkK | ||
| − | |||
==Video Solution 1 by SpreadTheMathLove== | ==Video Solution 1 by SpreadTheMathLove== | ||
Latest revision as of 20:01, 21 February 2025
Contents
Problem
Five distinct integers from 
to 
are chosen, and five distinct integers from 
to 
are chosen. No two numbers differ by exactly . What is the sum of the ten chosen numbers?
Solution
Note that for no two numbers to differ by , every number chosen must have a different units digit. To make computations easier, we can choose 
from the first group and 
from the second group. Then the sum evaluates to 
.
~Soupboy0
~ Edited by Aoum
Another Way To Compute
For 
, we can add the first term and the last term, which is 
. If we add the second term and the second-to-last term it is also . There are 
pairs that sum to , so the answer is 
which equals 
.
- leafy
Similar solution
One efficient method is to quickly add 
, which is 
. Then because you took 
in total away from , you add . 
.
~Bepin999
Video Solution by Pi Academy
https://youtu.be/Iv_a3Rz725w?si=E0SI_h1XT8msWgkK
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=jTTcscvcQmI
Video Solution (A Clever Explanation You’ll Get Instantly)
https://youtu.be/VP7g-s8akMY?si=DtG8sG4CK4RrUlVz&t=1815 ~hsnacademy
Video Solution by Thinking Feet
See Also
| 2025 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
