2013 AMC 10B Problems/Problem 24
Contents
Problem
A positive integer is nice if there is a positive integer
with exactly four positive divisors (including
and
) such that the sum of the four divisors is equal to
. How many numbers in the set
are nice?
Solution 1
A positive integer with only four positive divisors has its prime factorization in the form of , where
and
are both prime positive integers or
where
is a prime. One can easily deduce that none of the numbers are even near a cube so that case is finished. We now look at the case of
. The four factors of this number would be
,
,
, and
. The sum of these would be
, which can be factored into the form
. Easily we can see that now we can take cases again.
Case 1: Either or
is 2.
If this is true then we have to have that one of or
is odd and that one is 3. The other is still even. So we have that in this case the only numbers that work are even multiples of 3 which are 2010 and 2016. So we just have to check if either
or
is a prime. We see that in this case none of them work.
Case 2: Both and
are odd primes.
This implies that both and
are even which implies that in this case the number must be divisible by
. This leaves only
and
.
so either
or
both has a factor of
or one has a factor of
. If it was the first case, then
or
will equal
. That means that either
or
has a factor of
. That means that
or
is
which isn't a prime, so 2012 does not work.
so we have
. 503 and 3 are both odd primes, so 2016 is a solution. Thus the answer is
.
Solution 2
If has four divisors, then its divisors would be 1,
,
and
, where
and
are prime. Therefore, the sum of the divisors of
is
.
If either or
are odd, then
or
are even. Therefore,
and
are even, so
is a multiple of 4. The only two numbers from the range
that are multiples of 4 are 2012 and 2016.
Factoring 2012, we get . To make
and
even,
, we have
and
. However, if
was 1, then
is not prime, so 2012 is not nice.
Factoring 2016, we get .
, we have
.
Testing for the lowest , we get
and
. Therefore,
, and
, so
is nice, with
. Therefore, the answer is
.
See also
2013 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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