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 share this property?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
First note that the only number between and that shares this property is .
Consider now the numbers in the range to . There is exactly number, , that shares the property when the units digits is . There are exactly numbers, and , when the units digit is ; exactly numbers, , and , when the units digits is , and so on. Because the thousands digit is always , when the units digit is , for , the sum of the hundreds and tens digits must be . There are exactly ways for this to occur. Hence there are exactly
numbers that share this property.
The problems on this page are the property of the MAA's American Mathematics Competitions