2022 AMC 10B Problems/Problem 8
- The following problem is from both the 2022 AMC 10B #8 and 2022 AMC 12B #6, so both problems redirect to this page.
Contents
Problem
Consider the following sets of
elements each:
How many of these sets contain exactly two multiples of
?
Solution 1
We apply casework to this problem. The only sets that contain two multiples of seven are those for which:
- The multiples of
are
and
That is, the first and eighth elements of such sets are multiples of
- The multiples of
are
and
That is, the second and ninth elements of such sets are multiples of
- The multiples of
are
and
That is, the third and tenth elements of such sets are multiples of
The first element is for some integer
It is a multiple of
when
The second element is for some integer
It is a multiple of
when
The third element is for some integer
It is a multiple of
when
Each case has sets. Therefore, the answer is
~MRENTHUSIASM
Solution 2
We find a pattern.
Through quick listing
, we can figure out that the first set has
multiple of
. The second set has
multiple of
. The third set has
multiples of
. The fourth set has
multiple of
. The fifth set has
multiples of
. The sixth set has
multiple of
. The seventh set has
multiples of
. The eighth set has
multiple of
. The ninth set has
multiples of
. The tenth set has
multiples of
.
We see that the pattern for the number of multiples per set goes:
We can reasonably conclude that the pattern
repeats every
times. So, for every
sets, there are three multiples of
. We calculate
and multiply that by
(We disregard the remainder of
since it doesn't add any extra sets with
multiples of
.). We get
.
Solution 3 (Fastest)
Each set contains exactly or
multiples of
.
There are total sets and
multiples of
.
Thus, there are sets with
multiples of
.
~BrandonZhang202415
Video Solution 1
~Education, the Study of Everything
Video Solution(1-16)
~~Hayabusa1
Video Solution by Interstigation
https://youtu.be/_KNR0JV5rdI?t=884 ~~(chimpionboy)(slight error) Can someone fix the pdf, it shows answer choices A and B as 42
See Also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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 | ||
All AMC 10 Problems and Solutions |
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
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 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.