If x is the smallesr positive integer such that x/2 is a perfect
square, x/3 is a perfect cube, and x/5 is a pefect fifth, what is the
largest power of 30 which divides x? (The method is more important
than the answer.)

let's assume that x is equal to (2^a)*(3^b)*(5^c).

(we know it's divisible by 2, 3, and 5, of course, and if it were also
divisible by 7, say, it would have to be bigger.)

so if x/2 is a perfect square, that means that 

I:
a-1 is even
b is even
c is even 

x/3 is a cube tells us that
II:
a is a multiple of 3
b-1 is a multiple of 3
c is a multiple of 3

x/5 is a fifth-power tells us
III:
a is a multiple of 5
b is a multiple of 5
c-1 is a multiple of 5

so we know from (II, III) that a is a multiple of 3 and 5, and thus a
multiple of 15, and since 15 satisfies (I), we can take a=15.

what numbers work for b?  looking at multiples of 5....
5 doesn't work (5-1 not a multiple of 3, violates II)
10 works (10-1 is a multiple of 3, and it's even)
so b=10

for c, c-1 has to be a multiple of 5, so possible c's are 1, 6, 11, 16, etc.

1 doesn't work, since it's not even
6 works!  it's even, and a multiple of 3.

so a=15 b= 10 c=6, and thus 

x= 2^15 * 3^10 * 5^6 

searching google for "2^15*3^10*5^6" tells us that x=3.0233088 × 10^13

30 is 2*3*5, of course, so 30^n will divide x if and only if x <= a, b, and c.

so n = 6.

in some ways, this is of course a trick question, but to solve it you
have to understand some interesting things about how composite numbers
are built from their prime divisors.

by searching google for "prime factorization elementary" I found some
good web pages about the topic, see for instance

http://mcraefamily.com/MathHelp/BasicNumberPrimeFactorization.htm

http://mathforum.org/library/drmath/view/58534.html
the whole Dr. Math website is a great resource for the kinds of
questions you're asking, see

http://mathforum.org/library/drmath/sets/mid_factornumb.html

for all of their questions/answers about primes and factors.

Hope this helps!

--David

Thanks especially for the links.