Difference between revisions of "2021 Fall AMC 10A Problems/Problem 8"
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Revision as of 16:36, 23 November 2021
A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?
Solution 1
Note that the number 
By the problem statement, ![\[10x + y = x + y^2 \Rightarrow 9x = y^2 - y \Rightarrow 9x = y(y-1).\]](http://latex.artofproblemsolving.com/8/7/f/87f4b391e5cf99c357f248a4a6d594e6545870c3.png)
From this we see that 
must be divisible by 
This only happens when 
Then, 
Thus, there is only 
cuddly number which is 
Thus, the answer is 
~NH14
See Also
| 2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 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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