2022 AMC 10A Problems/Problem 7
Contents
Problem
The least common multiple of a positive divisor 
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 that 
Therefore, we have 
The sum of its digits is 
~MRENTHUSIASM
Solution 2
Since the 
contains only factors of 
, 
, and 
, cannot be divisible by any other prime.
Let n = 
, where 
,
, and 
are nonnegative integers.
We know that 
= 
= 
=
= = 
Thus
(1)=
so
= (2)=
<=
<=
= 
<= (3)
From the gcf information, 
= gcf(n, 
) = 
= 
This means, that since 
, 
, so 
= 
and 
.
Hence, multiplying using 
, , gives 
and the sum of digits is hence $\boxed{\textbf{(B)} ~6}
~USAMO333
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 | |
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