2014 AMC 8 Problem 22 Number Theory

Understand the 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? $\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad \textbf{(E) }9$
Source of the problem
2014 AMC 8 problem 22
Topic
Number Theory
Difficulty Level
Easy
Suggested Book
Mathematical Circles

Start with hints

Do you really need a hint? Try it first!

Proceed by assuming the number to be \( 10a + b \ where \ a \ and \ b \ are  \ the \ digits \) .
Since the number is equal to the product of the digits \( a \times b \)  plus the sum of the digits \( (a + b) \)  .  Arrange an equation and proceed .
So \( 10a + b =  (a \times b) + (a + b)  \) .   Simplify and find the unit digit  i.e. \( b \)  .
Simplifying \( 10  \times a = (b+ 1) \times a \) . Dividing by \( a \), we have that \( b+ 1 = 10 \) . Therefore, the unit digit, \( b \) , is  9 .  

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