2005 AMC 10A Problems/Problem 16
Contents
Problem
The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is . How many two-digit numbers have this property?
Solution 1
Let the number be where and are the tens and units digits of the number.
So must have a units digit of
This is only possible if , so is the only way this can be true.
So the numbers that have this property are .
Therefore the answer is
Solution 2
Let a two-digit number equal , where and are the tens and units digits of the number.
From the problem, we have
Now let , where and are the tens and units digits of the number. Then it must be that as stated in the problem.
Note that ends in , but ends in , so . We need not to care about , since it cancels out in the calculation.
So the answer is , since there are numbers that have .
~BurpSuite
See Also
2005 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.