Difference between revisions of "2024 AMC 10B Problems/Problem 25"
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==Solution 1== | ==Solution 1== | ||
| − | The <math> | + | The <math>3</math>x<math>3</math>x<math>3</math> block has side lengths of <math>3a, 3b, 3c</math>. The <math>2</math>x<math>2</math>x<math>7</math> block has side lengths of <math>2b, 2c, 7a</math>. |
| − | We can create the following system of equations, knowing that the new block has 1 unit taller, deeper, and wider than the original: | + | We can create the following system of equations, knowing that the new block has <math>1</math> unit taller, deeper, and wider than the original: |
<cmath>3a+1 = 2b</cmath> | <cmath>3a+1 = 2b</cmath> | ||
<cmath>3b+1=2c</cmath> | <cmath>3b+1=2c</cmath> | ||
Revision as of 00:28, 14 November 2024
Solution 1
The 
xx block has side lengths of 
. The 
xx
block has side lengths of 
.
We can create the following system of equations, knowing that the new block has 
unit taller, deeper, and wider than the original:
![\[3a+1 = 2b\]](http://latex.artofproblemsolving.com/3/1/3/313d38b3f4e9711c2c6d9176d8b2f695b126ca2a.png)
![\[3b+1=2c\]](http://latex.artofproblemsolving.com/f/0/6/f064fed198a80e2d4df1412c2d682645ca4c5d37.png)
![\[3c+1=7a\]](http://latex.artofproblemsolving.com/c/8/2/c82d94c0eb58f9202d41b78c5a585a2e52b3abc5.png)
Adding all the equations together, we get 
. Adding 
to both sides, we get 
. The question states that 
are all relatively prime positive integers. Therefore, our answer must be congruent to 
. The only answer choice satisfying this is 
.
~lprado
