Question 1209726: Which is the result of multiplying all positive divisors of 2016?
a) 2016^19
b) 2016^16
c) 2016^18
d) 2016^17
e) 2016^15
Answer by math_tutor2020(3816)   (Show Source):
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Answer: 2016^18 which is choice C
Explanation
2016 = 2^5*3^2*7
d(n) = number of divisors of n
a,b,c = positive integers
p,q,r = primes
d(p^a*q^b*r^c) = (a+1)*(b+1)*(c+1) .... see formula (3) of this page and see this page
d(2^5*3^2*7^1) = (5+1)*(2+1)*(1+1)
d(2016) = 36
The value 2016 has 36 divisors.
Verification using WolframAlpha
There are many other online calculators that will perform a similar function.
The divisors pair up to multiply to 2016
Eg: 2*1008 = 2016 and 8*252 = 2016
Those 36 divisors form 36/2 = 18 copies of 2016 multiplied together when multiplying all the positive divisors.
Therefore we arrive at the answer 2016^18
This massive number is approximately 3.0257 * 10^59
It is roughly 30257 followed by 55 copies of 0.
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