2019 AMC 10B Problems/Problem 19
- The following problem is from both the 2019 AMC 10B #19 and 2019 AMC 12B #14, so both problems redirect to this page.
Problem
@Letbe the set of all positive integer divisors of
How many numbers are the product of two distinct elements of
How many numbers are the product of two distinct elements of 
@
Solution
The prime factorization of 
is 
. Thus, we choose two numbers 
and 
where 
and 
, whose product is 
, where 
and 
.
Notice that this is analogous to choosing a divisor of 
, which has 
divisors. However, some of the divisors of 
cannot be written as a product of two distinct divisors of , namely: 
, , 
, and 
. The last two cannot be so written because the maximum factor of containing only 
s or 
s (and not both) is only 
or 
. Since the factors chosen must be distinct, the last two numbers cannot be so written because they would require 
or 
. This gives 
candidate numbers. It is not too hard to show that every number of the form 
, where 
, and 
are not both 
or 
, can be written as a product of two distinct elements in 
. Hence the answer is 
.
See Also
| 2019 AMC 10B (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 | ||
| All AMC 10 Problems and Solutions | ||
| 2019 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 13 |
Followed by Problem 15 |
| 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. 
