2005 AMC 10A Problems/Problem 16

Problem

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is $6$ . How many two-digit numbers have this property?

$\mathrm{(A) \ } 5\qquad \mathrm{(B) \ } 7\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 19$

Solution

Let the number be $10a+b$ where $a$ and $b$ are the tens and units digits of the number.

So $(10a+b)-(a+b)=9a$ must have a units digit of $6$

This is only possible if $9a=36$ , so $a=4$ is the only way this can be true.

So the numbers that have this property are $40$ , $41$ , $42$ , $43$ , $44$ , $45$ , $46$ , $47$ , $48$ , $49$ .

Therefore the answer is $10\Rightarrow D$

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2005 AMC 10A Problems/Problem 16

Problem

Solution

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