2020 AMC 8 Problems/Problem 19
Problem
A number is called flippy if its digits alternate between two distinct digits. For example, 
and 
are flippy, but 
and 
are not. How many five-digit flippy numbers are divisible by 
Solution 1
A number is divisible by 
precisely if it is divisible by 
and 
. The latter means the last digit must be either or 
, and the former means the sum of the digits must be divisible by . If the last digit is , the first digit would be (because the digits alternate), which is not possible. Hence the last digit must be , and the number is of the form 
. If the unknown digit is 
, we deduce 
. We know 
exists modulo because 2 is relatively prime to 3, so we may conclude that (i.e. the second and fourth digit of the number) must be a multiple of . It can be , , 
, or 
, so there are 
options: 
, 
, 
, and 
.
Solution 2 (variant of Solution 1)
As in Solution 1, we find that such numbers must start with and alternate with (i.e. must be of the form ), where the two digits between the s need to be the same. Call that digit . For the number to be divisible by , the sum of the digits must be divisible by ; since the sum of the three s is , which is already a multiple of , it must also be the case that 
is a multiple of . Thus, the problem reduces to finding the number of digits from to for which 
is a multiple of . This leads to 
, , , or , so there are possible numbers (namely , , , and ).
See also
| 2020 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
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| All AJHSME/AMC 8 Problems and Solutions | ||
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