Let n be a positive integer such that the number of positive factors

Question

Let n be a positive integer such that the number of positive factors of  5^n  is 15. What is the number of positive factors of  9^n ?

A. 14
B. 15
C. 28
D. 29
E. 30

chetan2u · Answer

Number of positive factors of  5^n  is n+1 as ‘5’ is prime. 
Thus, n+1=15 or n=14

Get  9^n  in prime factorisation =>  9^n=3^{2n} 
Thus, number of positive factors = (2n+1) = 2*14+1 =29

rey1009 · Answer

Number of factors = n+1 =15
therefore n =14

for 9^n
prime factorisation is 3^2
9^n =3^2n
number of factors is 2n+1 =28+1= 29

Kinshook · Answer

Given: Let n be a positive integer such that the number of positive factors of  5^n  is 15. 
Asked: What is the number of positive factors of  9^n ?

Since the number of positive factors of  5^n  is 15. 
n=14

Number of positive factors of  9^n = 3^{2n} = 3^{28} = 29

IMO D

Mgerman42 · Answer

why is n 14? 15 5's would be 5^15. Clearly I'm wrong but the three answers to these questions all just list the same info

Krunaal · Answer

The question says that the  number of factors  of  5^{n}  is 15 and not the number of 5’s

To calculate number of factors of any integer,

First make prime factorization of an integer  k = a^{p} * b^{q} * c^{r} , where a, b, and c are prime factors of k and p, q, and r are their powers.

The number of factors of k will be expressed by the formula (p+1)(q+1)(r+1).

Example: Finding the number of all factors of 450:  2^{1} 3^{2} 5^{2}

Total number of factors of 450 including 1 and 450 itself is (1+1) * (2+1) * (2+1) = 18 factors.

In the given question k =  5^{n}  => we see 5 is already a prime number, for k to have 15 factors, n should be 14.

egmat · Answer

Your thinking:  "15 factors means 5^15"
 This mixes up the number of factors with the exponent.

When you have  5^n , the factors are:
5^0, 5^1, 5^2, 5^3, ... , 5^n

Notice we start from  5^0 = 1  (yes, 1 is always a factor!)

So if n =  14 , the factors are:
5^0, 5^1, 5^2, 5^3, ... , 5^14

Count them: that's positions 0 through 14, which gives us  15  factors.

Formula: Number of factors of p^k = k + 1

The "+1" comes from including p^0 = 1 as a factor.

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