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2005 AMC 10A Problem 16

Problem 16 of 25IntermediateNumber Theory

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is 6.6. How many two-digit numbers have this property?

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Solution

If the number is 10a+b,10a + b, then (10a+b)−(a+b)=9a.(10a + b) - (a + b) = 9a. The units digit of 9a9a is 66 only when a=4,a = 4, since 9⋅4=36.9 \cdot 4 = 36. The digit bb can then be anything from 00 to 9,9, giving the ten numbers 4040 through 49.49. Thus, the correct answer is D.

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Concepts: digits · place value · units digit

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.