If n is a positive integer such that 2n has 28 positive divisors and 3

Question

If n is a positive integer such that 2n has 28 positive divisors and 3n has 30 positive divisors, then how many positive divisors does 6n have?

(A) 32

(B) 34

(C) 35

(D) 36

(E) 38

nick1816 · Answer

Comparing to 2n, in 3n there is decrease in the exponent of 2 by 1 and increase in the the exponent of 3 by 1
28=1*14, 2*14, 4*7  or 2*2*7
30= 1*30, 2*15, 3*10, 5*6  or 5*3*2
Hence n should be equals to 2^5 *3^3
6n = 2^6 * 3^4
Numbers of factor of 6n= 7*5=35

neha283 · Answer

Can someone post a detailed solution and approach for this question?
TIA!

SiffyB · Answer

Hi  nick1816 ,

Could you explain the solution in more detail? Didn't quite get the method you used. Thanks!

nick1816 · Answer

2n = 2^p *a^q * b^r*.......*3^z ; Number of factors = (p+1)*(q+1)*......*(z+1)

3n = 2^{p-1} *a^q * b^r*.......*3^{z+1} ; Number of factors = (p)*(q+1)*......*(z+2)

In 3n, the exponent of 2 is reduced by 1 and exponent of 3 is increased by 1.

So i factor 28 and 30 in all possible ways and choose the combination in which one factor is reduced by 1 and other is increased by 1.

28= 4*7; 30= 5*6

4 +1 = 5 and 7-1 =6

Other way to solve the question

28=1*28, 2*14, 4*7 or 2*2*7

Case 1 - 28=1*28

2n = a^{27} , a is prime number; a must be equal to 2, as 2n has only 1 prime factor.

3n = 3*2^{26} . Number of factors = 2*27 = 54 ( Rejected )

Case 2-  28=2*14

2n = a*b^{13} . Either a= 2 or b=2

1)  2n = 2*b^{13} ;  3n = 3*b^{13}

number of factors = 28 , if b is not equal to 3 ( Rejected )

number of factors =  15  , if b is equal to 3 ( Rejected )

2)  2n = a*2^{13} ;  3n = 3*a*2^{12}

number of factors = 2*2*13 =  52 , if a is not equal to 3 ( Rejected )

number of factors = 3*13 =  39 , if a is equal to 3 ( Rejected )

Case 3-  28=4*7

2n = a^3*b^6 . Either a= 2 or b=2

1)  2n = 2^3*b^6 ;  3n = 3*2^2*b^6

number of factors = 2*3*7 =  42  , if b is not equal to 3 ( Rejected )

number of factors = 3*8= 24  , if b is equal to 3 ( Rejected )

2)  2n = a^3*2^6 ;  3n = 3*a^3*2^5

number of factors = 2*4*6 =  48  , if a is not equal to 3 ( Rejected )

number of factors = 5*6= 30 , if a is equal to 3 ( What we looking for)

SiffyB · Answer

Thank you for such a detailed explanation!! Helped me understand this so much better

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If n is a positive integer such that 2n has 28 positive divisors and 3

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