2014 AMC 8

Complete problem set with solutions and individual problem pages

Problem 22 Hard

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?

  • A.

    1

  • B.

    3

  • C.

    5

  • D.

    7

  • E.

    9

Answer:E

We can think of the number as 10a+b, where a is the tens digit and b is the unit digit. 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, which factors 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}

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