2000 AIME II Problems/Problem 4
Problem
What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?
Solution
We use the fact that the number of divisors of a number is
. If a number has
factors, then it can have at most
distinct primes in its factorization.
Dividing the greatest power of from
, we have an odd integer with six positive divisors, which indicates that it either is (
) a prime raised to the
th power, or two primes, one of which is squared. The smallest example of the former is
, while the smallest example of the latter is
.
Suppose we now divide all of the odd factors from ; then we require a power of
with
factors, namely
. Thus, our answer is
.
See also
2000 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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