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Increase in number of factors. (Posted on 2010-10-05) Difficulty: 2 of 5
If n has 15 factors (1 and n inclusive ) and 2n has 20 factors. What is the number of factors of 4n?

Also, what can be the number of factors of 5n ?

See The Solution Submitted by Vishal Gupta    
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Solution | Comment 2 of 3 |
n must be of the form p^14 or p^4*q^2 where p and q are distinct primes.  [The number if factors the product of the powers each increased by 1, (4+1)(2+1)=15]

Multiply each case by 2

In the first case if p is 2 we have 2^15 which has 16 factors
and if p is not 2 we have 2*p^14 which has 30 factors.

In the second case if niether p nor q is 2 we have 2*p^4*q^2 which has 30 factors
if p is 2 we have 2^5*q^2 which has 18 factors
if q is 2 we have p^4*2^3 which has 20 factors

So n is of the form 2^2*p^4,
4n is of the form 2^4*p^4 which has 25 factors

for 5n it depends on whether p is 5
if p is not 5 we have 2^2*p^4*5 which has 30 factors
if p is 5 we have 2^2*p^5 which has 18 factors

  Posted by Jer on 2010-10-05 14:55:03
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