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loanpt0101
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If n is a positive integer such that 2^51 and (2^n)(3^3) have the same 22 Dec 2023, 07:14
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If n is a positive integer such that \(2^{51}\) and \((2^n)(3^3)\) have the same number of positive factors, what is the value of n?

A. 12
B. 17
C. 28
D. 34
E. 48

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Bunuel
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If n is a positive integer such that 2^51 and (2^n)(3^3) have the same 22 Dec 2023, 07:18
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loanpt0101
If n is a positive integer such that \(2^{51}\) and \((2^n)(3^3)\) have the same number of positive factors, what is the value of n?

A. 12
B. 17
C. 28
D. 34
E. 48
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\(2^{51}\) has 51 + 1 = 52 factors.
\((2^n)(3^3)\) has (n + 1)(3 + 1) factors.

Hence, we are given that (n + 1)(3 + 1) = 52, which results in n = 12.

Answer: A.
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If n is a positive integer such that 2^51 and (2^n)(3^3) have the same 23 Dec 2023, 11:10
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loanpt0101

Asked: If n is a positive integer such that \(2^{51}\) and \((2^n)(3^3)\) have the same number of positive factors, what is the value of n?

Number of positive factors of 2^{51} = 52
Number of positive factors of 2^n*3^3 = (n+1)4

4(n+1) = 52
n+1 = 52/4 = 13
n = 12

IMO A
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If n is a positive integer such that 2^51 and (2^n)(3^3) have the same 26 Sep 2025, 03:54
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we add 1 to the number of factors because 1 is a common factor which we have to consider?
Bunuel


\(2^{51}\) has 51 + 1 = 52 factors.
\((2^n)(3^3)\) has (n + 1)(3 + 1) factors.

Hence, we are given that (n + 1)(3 + 1) = 52, which results in n = 12.

Answer: A.
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If n is a positive integer such that 2^51 and (2^n)(3^3) have the same 26 Sep 2025, 03:58
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we add 1 to the number of factors because 1 is a common factor which we have to consider?

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No.

Finding the Number of Factors of an Integer

First, make the 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 respective 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: \(450 = 2^1 * 3^2 * 5^2\)
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If n is a positive integer such that 2^51 and (2^n)(3^3) have the same 27 Sep 2025, 01:30
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Hi Pikachu9,

Good question:
You need to understand the formula.
Going back to basics (Trying my best to explain):

Consider Factors of 6 = 1,2,3 & 6
Write the prime factor of 6 = 2 and 3
Basically: 6 = 2*3
6 = (2^0*2^1)(3 ^ 0*3^1)

Now for rule of factorization:
1st factor: 2^0*3 ^0 = 1
2nd factor: 2^0*3 ^1 = 3
3rd factor: 2^1*3 ^0 = 2
4th factor: 2^1*3 ^1 = 6

No of factors: 4
By direct formula :
=2^1 *3 ^1
= (1+1)*(1+1)
= 2*2
= 4
Hope I was able to make it clear.
pikachu9
we add 1 to the number of factors because 1 is a common factor which we have to consider?

Show more
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