Difference between revisions of "2013 AMC 10B Problems/Problem 18"
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==Problem== | ==Problem== | ||
| − | The number <math>2013</math> has the property that its units digit is the sum of its other digits, that is <math>2+0+1=3</math>. How many integers less than <math>2013</math> but greater than <math>1000</math> | + | The number <math>2013</math> has the property that its units digit is the sum of its other digits, that is <math>2+0+1=3</math>. How many integers less than <math>2013</math> but greater than <math>1000</math> have this property? |
<math> \textbf{(A)}\ 33\qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ 58 </math> | <math> \textbf{(A)}\ 33\qquad\textbf{(B)}\ 34\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ 58 </math> | ||
Revision as of 14:52, 20 August 2020
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
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 
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
