Difference between revisions of "2015 AMC 12A Problems/Problem 10"
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==Solution 2== | ==Solution 2== | ||
| − | Plug in values of <math>x</math> and solve for <math>y</math>, noting <math>x > y</math> and that y is an integer. | + | Plug in values of <math>x</math> and solve for <math>y</math>, noting <math>x > y</math> and that <math>y</math> is an integer. |
<math>x = 8</math>: | <math>x = 8</math>: | ||
| Line 19: | Line 19: | ||
8 + y + 8y = 80 | 8 + y + 8y = 80 | ||
</math> | </math> | ||
| − | |||
<math>y = 8</math> (Does not work because <math>x > y</math>) | <math>y = 8</math> (Does not work because <math>x > y</math>) | ||
| Line 30: | Line 29: | ||
</math> | </math> | ||
| − | |||
<math>y = 70/11</math> (Does not work because <math>y</math> is not an integer) | <math>y = 70/11</math> (Does not work because <math>y</math> is not an integer) | ||
| Line 39: | Line 37: | ||
15 + y + 15y = 80 | 15 + y + 15y = 80 | ||
</math> | </math> | ||
| − | |||
<math>y = 65/16</math> (Does not work because <math>y</math> is not an integer) | <math>y = 65/16</math> (Does not work because <math>y</math> is not an integer) | ||
| Line 49: | Line 46: | ||
18 + y + 18y = 80 | 18 + y + 18y = 80 | ||
</math> | </math> | ||
| − | |||
<math>y = 62/19</math> (Does not work because <math>y</math> is not an integer) | <math>y = 62/19</math> (Does not work because <math>y</math> is not an integer) | ||
| − | Thus <math>x = 26</math>, <math> | + | Thus <math>x = 26</math>, <math>\boxed{\textbf{(E)}\ 26}</math>. |
Latest revision as of 15:47, 28 September 2025
Problem
Integers 
and 
with 
satisfy 
. What is ?
Solution 1
Use SFFT to get 
. The terms 
and 
must be factors of 
, which include 
. Because 
, 
is equal to 
or . But if 
, then 
and so 
.
Solution 2
Plug in values of and solve for , noting and that is an integer.
:
(Does not work because )
:
(Does not work because is not an integer)
:
(Does not work because is not an integer)
:
(Does not work because is not an integer)
Thus 
, 
.
~ Solution by CYB3RFLARE7408
Video Solution by OmegaLearn
https://youtu.be/ba6w1OhXqOQ?t=4512
~ pi_is_3.14
See Also
| 2015 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 9 |
Followed by Problem 11 |
| 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 12 Problems and Solutions | |
