Product of divisors number theory problem

[lei123]

Homework Statement

prove using induction:
for any n =1,2,3...
the product of the divisors of n = n^(number of divisors of n (counting 1 and n)/2)

Homework Equations

The Attempt at a Solution

I understand why this is the case, but I'm having trouble with the induction step.
if the product of the divisors of k = k^(number of divisors of k/2), the the product of the divisors of k+1 = k^(number of divisors of k+1/2). I know that k and k+1 are relatively prime, so all their divisors are different. But I can't seem to make that final connection

 

[tiny-tim]

hi lei123!

i've no idea why anyone would want to prove it by induction …

but if you do, then as you say, using induction on the number n itself won't work, so how about doing it on the number of divisors, or on the number of prime factors?