2020 AMC 8 Problems/Problem 19
Contents
Problem 19
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 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
).
Solution 3 (Variant of Solution 1, but using Brute Force)
After finding out that the last digit must be , the number is of the form
. If the unknown digit is
, we can find that one of the solutions to
is
, since
is equal to
, which is divisible by
. After trying every one digit number, you'll notice that
must be a multiple of
, meaning that
,
,
, or
.
,
,
, and
are the
solutions to this question.
Solution 4 (mods)
assume the number is
Solutions:
Video Solution by North America Math Contest Go Go Go
https://www.youtube.com/watch?v=5Qo4pG3Uk_U
~North America Math Contest Go Go Go
Video Solution by WhyMath
~savannahsolver
Video Solution
Video Solution by Interstigation
https://youtu.be/YnwkBZTv5Fw?t=980
~Interstigation
See also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
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