2000 AMC 8 Problems/Problem 11
Problem
The number has the property that it is divisible by its unit digit. How many whole numbers between 10 and 50 have this property?
Solution
Casework by the units digit will help organize the answer.
gives no solutions, since no real numbers are divisible by
has solutions, since all numbers are divisible by .
has solutions, since every number ending in is even (ie divisible by ).
has solution: . or will retain the units digit, but will stop the number from being divisible by . is the smallest multiple of that will keep the number divisible by , but those numbers are and , which are out of the range of the problem.
has solutions: and . Adding or subtracting will kill divisibility by , since is not divisible by .
has solutions: every number ending in is divisible by .
has solution: . or will kill divisibility by , and thus kill divisibility by .
has no solutions. The first multiples of that end in are and , but both are outside of the range of this problem.
has solution: . will all kill divisibility by since and are not divisible by .
has no solutions. and are the smallest multiples of that end in .
Totalling the solutions, we have solutions, giving the answer , which is 17.
See Also
2000 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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