Difference between revisions of "2014 AMC 8 Problems/Problem 22"

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==Solution==
 
==Solution==
 
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We can think of the number as <math>10a+b</math>, where a and b are digits. Since the number is equal to the product of the digits (<math>a\cdot b</math>) plus the sum of the digits (<math>a+b</math>), we can say that <math>10a+b=a\cdot b+a+b</math>. We can simplify this to <math>10a=a\cdot b+a</math>, and factor to <math>(10)a=(b+1)a</math>. Dividing by <math>a</math>, we have that <math>b+1=10</math>. Therefore, the units digit, <math>b</math>, is <math>\textbf{(E) }9</math>
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2014|num-b=21|num-a=23}}
 
{{AMC8 box|year=2014|num-b=21|num-a=23}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 21:46, 26 November 2014

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 ($a\cdot b$) plus the sum of the digits ($a+b$), we can say that $10a+b=a\cdot b+a+b$. We can simplify this to $10a=a\cdot b+a$, and factor to $(10)a=(b+1)a$. Dividing by $a$, we have that $b+1=10$. Therefore, the units digit, $b$, is $\textbf{(E) }9$

See Also

2014 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 21
Followed by
Problem 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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