# 2014 AMC 8 Problems/Problem 22

## Problem

A $2$-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number? $\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad \textbf{(E) }9$

## Solution

We can think of the number as $10a+b$, where a and b are digits. Since the number is equal to the product of the digits ( $ab$) plus the sum of the digits ( $a+b$), we can say that $10a+b=ab+a+b$. We can simplify this to $10a=ab+a$, and factor to $(10)a=(b+1)a$. Dividing by $a$, we have that $b+1=10$. Therefore, the units digit, $b$, is $\boxed{\textbf{(E) }9}$

## Solution 2

A two digit number is namely $10a+b$, where $a$ and $b$ are digits in which $0 < a \leq 9$ and $0 \leq b \leq 9$. Therefore, we can make an equation with this information. We obtain $10a+b=(a \cdot b) + (a + b)$. This is just $10a+b=ab+a+b.$ Moving $a$ and $b$ to the right side, we get $9a=ab.$ Cancelling out the $a$s, we get $9=b$ which is our desired answer as $b$ is the second digit. Thus the answer is $\boxed{\textbf{(E)}9}$. ~mathboy282

## See Also

 2014 AMC 8 (Problems • Answer Key • Resources) Preceded byProblem 21 Followed byProblem 23 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 AJHSME/AMC 8 Problems and Solutions

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