2013 AMC 10B Problems/Problem 18
Problem
The number 
has the property that its units digit is the sum of its other digits, that is 
. How many integers less than but greater than 
have this property?
Solution 1.1
We take cases on the thousands digit, which must be either 
or 
:
If the number is of the form 
where 
are digits, then we must have 
Since 
we must have 
By casework on the value of 
, we find that there are 
possible pairs 
, and each pair uniquely determines the value of 
, so we get 
numbers with the given property.
If the number is of the form 
then it must be one of the numbers 
Checking all these numbers, we find that only 
has the given property.
Therefore, the number of numbers with the property is 
Solution 1.2
This solution picks up from finding that 
in solution 1.1. Instead of using casework to find all possible pairs, , let's introduce a dummy variable, 
. Let us now have that 
, where 
are all nonnegative.
We may now use stars and bars to distribute units between 
and . Any units that is given will essentially be discarded - this is how we get the 'less than' in the 'less than or equal to 
' relation we found earlier.
Using two dividers, we find that the number of distributions is 
which is . We proceed from here as above.
Video Solution
~savannahsolver
See also
| 2013 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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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