Question

A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$

$3$

$4$

$5$

$6$

$8$

Answer: B

Concepts:divisibility digits

Difficulty rating: 1270

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Written solution:

A five-digit flippy number has the form $ABABA,$ where $A\ne0$ and $A\ne B.$ Divisibility by $5$ requires the last digit $A$ to be $0$ or $5.$ Since $A\ne0,$ we must have $A=5.$

The number therefore has the form $5B5B5.$ Its digit sum is $15+2B,$ so divisibility by $3$ requires $B$ to be a multiple of $3.$ The possibilities are $B=0,3,6,9,$ giving $4$ numbers.

Thus, the correct answer is B.

2020 AMC 8 Problem 19

Answer: B

Difficulty rating: 1270

Video solution:

Written solution:

Problem 19 in Other Years