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 
. How many two-digit numbers have this property?
Solution 1
Let the number be 
, where 
is its tens digit and 
is its units digit. Then 
must have a units digit of , and as is the tens digit, we can only have 
, so 
.
Accordingly, 
has units digit only if 
. Thus the numbers that have the required property are all those with tens digit 
, from 
to 
, so the answer is 
.
Solution 2
As in Solution 1, suppose that and are the tens and units digits of the number respectively, so the result of the subtraction is 
. Thus must have units digit .
We now observe that 
is a multiple of 10, so has units digit 
, and hence 
will have units digit 
(where the will become 
in the subtraction by 'borrowing' 
from the tens digit). Since is a single digit, this simply means 
, while can be any digit from to 
(since it cancelled out in the subtraction above).
Thus, as in Solution 1, there are possible choices for with , so the answer is 
.
~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 | ||
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