Difference between revisions of "2021 Fall AMC 10A Problems/Problem 8"

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~Steven Chen (www.professorchenedu.com)
 
~Steven Chen (www.professorchenedu.com)
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== Solution 3 ==
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If the tens digit is <math>a</math> and the ones digit is <math>b</math> then the number is <math>10+b</math> so we have the equation <math>10a + b = a + b^2</math>. We can guess and check after narrowing the possible cuddly numbers down to <math>13,14,24,25,35,36,46,47,57,68,78,89,</math> and <math>99</math>. (We can narrow it down to these by just thinking about how <math>a</math>'s value affects <math>b</math>'s value and then check all the possiblities.) Checking all of these we get that there is only <math>\boxed{1}</math> 2-digit cuddly number, and it is <math>89</math>.
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~Andlind
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2021 Fall|ab=A|num-b=7|num-a=9}}
 
{{AMC10 box|year=2021 Fall|ab=A|num-b=7|num-a=9}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 23:55, 25 November 2021

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 (ProblemsAnswer KeyResources)
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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