SOLUTION: what is product of the divisors of 1000 (including 1 and 1000)? a) 10^24 b) 10^16 c) 10^8 d) 10^30 e) 10^32

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Question 234483: what is product of the divisors of 1000 (including 1 and 1000)?
a) 10^24 b) 10^16 c) 10^8 d) 10^30 e) 10^32

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

1000+=+2%5E3%2A5%5E3 has only the prime divisors 2 and 5.
Every divisor of 1000 is therefore of the form 2%5Ep%2A5%5Eq,
where p and q are elements of {0,1,2,3}.  Since there
are 4 choices for p and 4 choices for q, there are 16
divisors of 1000.  That isn't necessary to know.  But
in the product of all 16 divisors of 1000, 2%5E0%2A2%5E1%2A2%5E2%2A2%5E3
or 2%5E6, and 5%5E0%2A5%5E1%2A5%5E2%2A5%5E3 or 5%5E6 occur exactly 4 times each.
Since 2%5E6 and 5%5E6 both occur 4 times each in the product
of all divisors, the product of the divisors must be
%282%5E6%29%5E4%2A%285%5E6%29%5E4+=+2%5E24%2A5%5E24+=+10%5E24, choice a)

Edwin