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2021 Fall AMC 10A Problems/Problem 8

Revision as of 00:55, 26 November 2021 by Andlind (talk | contribs)

Problem

A two-digit positive integer is said to be $\emph{cuddly}$ 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?

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Solution 1

Note that the number $\underline{xy} = 10x + y.$ By the problem statement, \[10x + y = x + y^2 \implies 9x = y^2 - y \implies 9x = y(y-1).\] From this we see that $y(y-1)$ must be divisible by $9.$ This only happens when $y=9.$ Then, $x=8.$ Thus, there is only $\boxed{\textbf{(B) }1}$ cuddly number, which is $89.$

~NH14

Solution 2

Denote this number as $\overline{ab}$.

Hence, we have $\overline{ab} = a + b^2$.

This can be written as $10 a + b = a + b^2$.

Hence, $b \left( b - 1 \right) = 9 a$. This implies $9 | b \left( b - 1 \right)$.

Hence, either $9 | b$ or $9 | b - 1$. Because $b \in \left\{ 0 , 1 , \cdots , 9 \right\}$, $b = 9$ or 1.

For $b = 9$, we get $a = 8$. This is a solution.

For $b = 1$, we get $a = 0$. However, recall that $a \neq 0$. Hence, this is not a solution.

Therefore, the answer is $\boxed{\textbf{(B) }1}$.

~Steven Chen (www.professorchenedu.com)

Solution 3

If the tens digit is $a$ and the ones digit is $b$ then the number is $10+b$ so we have the equation $10a + b = a + b^2$. We can guess and check after narrowing the possible cuddly numbers down to $13,14,24,25,35,36,46,47,57,68,78,89,$ and $99$. (We can narrow it down to these by just thinking about how $a$'s value affects $b$'s value and then check all the possiblities.) Checking all of these we get that there is only $\boxed{1}$ 2-digit cuddly number, and it is $89$.

~Andlind

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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