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

(Created page with "== Solution 1 == Note that the number <math>\overline{xy} = 10x + y.</math> By the problem statement, <cmath>10x + y = x + y^2 \Rightarrow 9x = y^2 - y \Rightarrow 9x = y(y-1)...")
 
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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?
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<math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4</math>
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== Solution 1 ==
 
== Solution 1 ==
 
Note that the number <math>\overline{xy} = 10x + y.</math> By the problem statement, <cmath>10x + y = x + y^2 \Rightarrow 9x = y^2 - y \Rightarrow 9x = y(y-1).</cmath> From this we see that <math>y(y-1)</math> must be divisible by <math>9.</math> This only happens when <math>y=9.</math> Then, <math>x=8.</math> Thus, there is only <math>1</math> cuddly number which is <math>89.</math> Thus, the answer is <math>\boxed{\textbf{(B).}}</math>
 
Note that the number <math>\overline{xy} = 10x + y.</math> By the problem statement, <cmath>10x + y = x + y^2 \Rightarrow 9x = y^2 - y \Rightarrow 9x = y(y-1).</cmath> From this we see that <math>y(y-1)</math> must be divisible by <math>9.</math> This only happens when <math>y=9.</math> Then, <math>x=8.</math> Thus, there is only <math>1</math> cuddly number which is <math>89.</math> Thus, the answer is <math>\boxed{\textbf{(B).}}</math>
  
 
~NH14
 
~NH14

Revision as of 19:42, 22 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?

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

Solution 1

Note that the number $\overline{xy} = 10x + y.$ By the problem statement, \[10x + y = x + y^2 \Rightarrow 9x = y^2 - y \Rightarrow 9x = y(y-1).\] 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 $1$ cuddly number which is $89.$ Thus, the answer is $\boxed{\textbf{(B).}}$

~NH14

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