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A -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?

2014 AMC 8 problem 22

Number Theory

Easy

Mathematical Circles

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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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