Difference between revisions of "2021 Fall AMC 10A Problems/Problem 8"
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== Problem == | == Problem == | ||
| − | 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? | + | A two-digit positive integer is said to be <math>\emph{cuddly}</math> 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? |
<math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4</math> | <math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4</math> | ||
Revision as of 18:42, 24 November 2021
Problem
A two-digit positive integer is said to be 
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
Note that the number 
By the problem statement, ![\[10x + y = x + y^2 \implies 9x = y^2 - y \implies 9x = y(y-1).\]](http://latex.artofproblemsolving.com/7/9/9/79916ece27f39c825bfa425031e66cc209c3ee04.png)
From this we see that 
must be divisible by 
This only happens when 
Then, 
Thus, there is only 
cuddly number, which 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
