2022 AMC 10A Problems/Problem 7
- The following problem is from both the 2022 AMC 10A #7 and 2022 AMC 12A #4, so both problems redirect to this page.
Problem
The least common multiple of a positive integer and
is
, and the greatest common divisor of
and
is
. What is the sum of the digits of
?
Solution 1
Note that
From the least common multiple condition, we conclude that
where
From the greatest common divisor condition, we conclude that
Therefore, we have The sum of its digits is
~MRENTHUSIASM
Solution 2
Since the LCM contains only factors of ,
, and
,
cannot be divisible by any other prime.
Let , where
,
, and
are nonnegative integers.
We know that
Thus,
(1) , so
.
(2) , so
.
(3) .
From the GCD information,
This means, that since
, it follows that
, so
and
.
Hence, multiplying using
,
,
gives
and the sum of digits is
.
~USAMO333
Video Solution 1 (Quick and Easy)
~Education, the Study of Everything
See Also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 3 |
Followed by Problem 5 |
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.