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
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.