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If the number of different positive factors of (2^y)(3^3) is the same

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Awli
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If the number of different positive factors of (2^y)(3^3) is the same [#permalink] New post   Updated on: 14 Apr 2015, 04:31
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If the number of different positive factors of (2^y)(3^3) is the same as the number of different factors of (2^51), what is the value of y?

A. 11
B. 12
C. 13
D. 48
E. 51
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B

Originally posted by Awli on 13 Apr 2015, 09:43.
Last edited by Bunuel on 14 Apr 2015, 04:31, edited 1 time in total.
Edited the question.
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Lucky2783
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Re: If the number of different positive factors of (2^y)(3^3) is the same [#permalink] New post   13 Apr 2015, 10:51
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Awli wrote:
If the number of different positive factors of (2^y)(3^3) is the same as the number of different factors of (2^51), what is the value of y?

a) 11

b) 12

c) 13

d) 48

e) 51


Number of factors of \(N= a^x * b^y\) where 'a' and 'b' are prime factors of N is

(X+1) (y+1)


So 4 (y+1)=52
y=12
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Re: If the number of different positive factors of (2^y)(3^3) is the same [#permalink] New post   13 Apr 2015, 23:38
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Its the number of factor .....
(x+1) (3+1) = 51+1------------------ equation no 1
(x+1)4 = 52
if you solve it will become
y =12

number of factors theory :
N = (a^n )(b^m)
a, b are prime factors of N raised to power n and m .

and number of factors = (n+1)(m+1) -------- this is what use in equation no 1

kudos appreciated . :)
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Re: If the number of different positive factors of (2^y)(3^3) is the same [#permalink] New post   14 Apr 2015, 04:30
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Awli wrote:
If the number of different positive factors of (2^y)(3^3) is the same as the number of different factors of (2^51), what is the value of y?

a) 11

b) 12

c) 13

d) 48

e) 51


Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

According to the above, the number of factors o f (2^y)(3^3) is (y + 1)(3 + 1) = 4y +4 and the number of factors of 2^51 is 51 + 1 = 52. Thus given that 4y +4 = 52 --> y = 12.

Answer: B.

Hope it's clear.
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Re: If the number of different positive factors of (2^y)(3^3) is the same [#permalink] New post   13 Apr 2015, 11:28
Lucky2783 wrote:
Awli wrote:
If the number of different positive factors of (2^y)(3^3) is the same as the number of different factors of (2^51), what is the value of y?

a) 11

b) 12

c) 13

d) 48

e) 51


Number of factors of \(N= a^x * b^y\) where 'a' and 'b' are prime factors of N is

(X+1) (y+1)


So 4 (y+1)=52
y=12



Why 4 (y+1) ??

Thanks
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Re: If the number of different positive factors of (2^y)(3^3) is the same [#permalink] New post   31 May 2023, 20:24
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Re: If the number of different positive factors of (2^y)(3^3) is the same [#permalink]
31 May 2023, 20:24
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