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
@
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 |
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