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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
Number Theory
Easy
Suggested Book
Mathematical Circles

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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