1996 AHSME Problems/Problem 29
Contents
Problem
If is a positive integer such that
has
positive divisors and
has
positive divisors, then how many positive divisors does
have?
Solution 1
Working with the second part of the problem first, we know that has
divisors. We try to find the various possible prime factorizations of
by splitting
into various products of
or
integers.
The variables are different prime factors, and one of them must be
. We now try to count the factors of
, to see which prime factorization is correct and has
factors.
In the first case, is the only possibility. This gives
, which has
factors, which is way too many.
In the second case, gives
. If
, then there are
factors, while if
, there are
factors.
In the second case, gives
. If
, then there are
factors, while if
, there are
factors.
In the third case, gives
. If
, then there are
factors, while if
, there are
factors.
In the third case, gives
. If
, then there are
factors, while if
, there are
factors.
In the fourth case, gives
. If
, then there are
factors. This is the factorization we want.
Thus, , which has
factors, and
, which has
factors.
In this case, , which has
factors, and the answer is
Solution 2
Because has
factors and
has
factors, we should rewrite the number
As the formula for the number of divisors for such a number gives:
We plug in the variations we need to make for the cases
and
.
has
has
If we take the top and divide by the bottom, we get the following equation:
. Letting
and
for convenience and expanding this out gives us:
We can use Simon's Favorite Factoring Trick (SFFT) to turn this back into:
or
As we want to be dealing with rather reasonable numbers for and
, we try to make the
term the slightly larger term and the
term the slightly smaller term. This effect is achieved when
and
. Therefore,
. We get that this already satisfies the requirements for the number we are looking for, and we take
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 28 |
Followed by Problem 30 | |
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 • 26 • 27 • 28 • 29 • 30 | ||
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