2000 AMC 8 竞赛试题/第11题
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 
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