Difference between revisions of "2005 AMC 10A Problems/Problem 16"
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The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is <math>6</math>. How many two-digit numbers have this property? | The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is <math>6</math>. How many two-digit numbers have this property? | ||
| − | <math> \ | + | <math> \textbf{(A) } 5\qquad \textbf{(B) } 7\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \textbf{(E) } 19 </math> |
==Solution== | ==Solution== | ||
Revision as of 18:29, 13 December 2021
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
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 
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. 
