Ain't that straightforward anymore

Number Theory Level 4

$\large 2015\times 2^5\times 3^4\times 5^6\times 7\times 11^6$

Determine the number of positive factors of the number above which are a multiple of $15$ .

The answer is 9408.

1 solution

Alex Zhong
Mar 30, 2015

$2015=5\times 13\times 31.$

Pull $3\times 5$ from the expression, we have the following left:

$2^5\times 3^3\times 5^6\times 7 \times 11^6\times 13 \times 31$

Number factors: $6\times 4\times 7\times 2\times 7\times 2\times 2 = \boxed{9408}.$

@Calvin Lin Though I've found the same solution, I think in the last step, 1 is also included as a divisor which is not divisible by 15. Hence, I would like you to review this problem's solution.

Kunal Verma - 6 years, 2 months ago

15 is a multiple of 15. I don't see any problem.

Alex Zhong - 6 years, 2 months ago

I'm not talking about 15 lol. I'm talking about 1. The formula:-

N = (p+1)(q+1)(r+1)

Where N is the number of divisors of

$A\quad =\quad { a }^{ p }{ b }^{ q }{ c }^{ r }$

Also includes 1 as a divisor.

In this case, we know 1 cannot be divided by 15.

Hence, I think the answer should be 9408 - 1 = 9407 and not 9408.

Please correct me if I'm going wrong. I do not intend to argue with anyone.

Kunal Verma - 6 years, 2 months ago

@Kunal Verma – 15 is already taken out from the expression .

Alex Zhong - 6 years, 2 months ago

@Alex Zhong – Ah so you say 15 fills up for the 1 subtracted?

Kunal Verma - 6 years, 2 months ago

@Kunal Verma – After 3 and 5 are taken out, the "1" means 15x1.

Kenny Lau - 5 years, 11 months ago

Ain't that straightforward anymore

The answer is 9408.

1 solution

0 pending reports