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nick13
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How many positive integer divisors does 8! have? 28 Oct 2023, 17:52
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How many positive integer divisors does 8! have?

A. 14
B. 15
C. 24
D. 56
E. 96

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How many positive integer divisors does 8! have? 28 Oct 2023, 23:33
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nick13
How many positive integer divisors does 8! have?

a. 14
b. 15
c. 24
d. 56
e. 95
Show more

Positive integers mean 1 and above. So the smallest is 1 and largest is 8!.

Now, the method to find positive integer divisors is as shown by gmatophobia
Get the number in its prime factors and take product of all exponents after adding one to each exponent.

Number of 2s: \(\frac{8}{2}+\frac{8}{2^2}+\frac{8}{2^3}\)=4+2+1=7
Number of 3s: \(\frac{8}{3}+\frac{8}{3^2}\)=2+0=2
Number of 5s and 7s will be 1 each.

Thus, 8!=\(2^7*3^2*5*7\) and factors are \((7+1)(2+1)(1+1)(1+1)=8*3*2*2=96\)

Clearly, Option E has been wrongly written as 95. Moreover, an odd number of positive factors means the number is Perfect Square and 8! is clearly not a perfect square.
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How many positive integer divisors does 8! have? 28 Oct 2023, 20:56
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nick13
How many positive integer divisors does 8! have?

a. 14
b. 15
c. 24
d. 56
e. 95
Show more

\(8! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 = 1 * 2 * 3 * 2^2 * 5 * (3 * 2) * 7 * 2^3 \)

\(= 2^7 * 3^2 * 5^1 * 7^1\)

Number of factors \(= 8 * 3 * 2 * 2 = 96\)
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How many positive integer divisors does 8! have? 25 Nov 2023, 13:22
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How many positive integer divisors does 8! have?

A) 14
B) 15
C) 24
D) 56
E) 96
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The prime number 8! consists are 2, 3, 5, and 7

  • Power of 2 in 8! = 7
  • Power of 3 in 8! = 2
  • Power of 5 in 8! = 1
  • Power of 7 in 8! = 1

\(8! = 2^7 * 3^2 * 5^1 * 7^1\)

Number of factors = \((7+1)*(2+1)*(1+1)*(1+1) = 8 * 3 * 2 * 2 = 12 * 8 = 96\)

Option E
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How many positive integer divisors does 8! have? 23 Dec 2023, 10:11
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can someone please explain how this to 2^7∗3^2∗5^1∗7^1 gets reduced to Number of factors being 8∗3∗2∗2=96
is there a specific formula or method to this ?
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How many positive integer divisors does 8! have? 23 Dec 2023, 10:26
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diya7
can someone please explain how this to 2^7∗3^2∗5^1∗7^1 gets reduced to Number of factors being 8∗3∗2∗2=96
is there a specific formula or method to this ?
Show more

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: \(450 = 2^1 * 3^2 * 5^2\)

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

2. Properties of Integers



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How many positive integer divisors does 8! have? 28 Jan 2024, 14:57
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Bunuel
diya7
can someone please explain how this to 2^7∗3^2∗5^1∗7^1 gets reduced to Number of factors being 8∗3∗2∗2=96
is there a specific formula or method to this ?

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: \(450 = 2^1 * 3^2 * 5^2\)

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

2. Properties of Integers



For other subjects:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.
Show more

Is this actually a formula we just have to keep in mind? like b^2+4ac
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How many positive integer divisors does 8! have? 28 Jan 2024, 15:00
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Bunuel
diya7
can someone please explain how this to 2^7∗3^2∗5^1∗7^1 gets reduced to Number of factors being 8∗3∗2∗2=96
is there a specific formula or method to this ?

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: \(450 = 2^1 * 3^2 * 5^2\)

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

2. Properties of Integers



For other subjects:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.

Is this actually a formula we just have to keep in mind? like b^2+4ac
Show more

Yes, knowing and understanding the formula for finding the number of factors can be useful for the GMAT.
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How many positive integer divisors does 8! have? 15 Nov 2024, 21:05
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can you help explain the intuition behind this rule?> i had no idea to use this.
(7+1)(2+1)(1+1)(1+1)=8∗3∗2∗2=96

chetan2u
nick13
How many positive integer divisors does 8! have?

a. 14
b. 15
c. 24
d. 56
e. 95

Positive integers mean 1 and above. So the smallest is 1 and largest is 8!.

Now, the method to find positive integer divisors is as shown by gmatophobia
Get the number in its prime factors and take product of all exponents after adding one to each exponent.

Number of 2s: \(\frac{8}{2}+\frac{8}{2^2}+\frac{8}{2^3}\)=4+2+1=7
Number of 3s: \(\frac{8}{3}+\frac{8}{3^2}\)=2+0=2
Number of 5s and 7s will be 1 each.

Thus, 8!=\(2^7*3^2*5*7\) and factors are \((7+1)(2+1)(1+1)(1+1)=8*3*2*2=96\)

Clearly, Option E has been wrongly written as 95. Moreover, an odd number of positive factors means the number is Perfect Square and 8! is clearly not a perfect square.
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How many positive integer divisors does 8! have? 02 Dec 2024, 18:56
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So, the idea is that we can have seven two's in a factor or we can have 0 2's ie one. Likewise for all numbers mentioned. Consider the number 12.
12's factors are: 1,2,3,4,6,12
It's prime factorization is: 2^2*3. So notice each factor can be represented in the following manner 3*2^0, 3*2^1, 3*2^2, 3^0*2, 3^0*2^0. I hope this is clear.
Dbrunik
can you help explain the intuition behind this rule?> i had no idea to use this.
(7+1)(2+1)(1+1)(1+1)=8∗3∗2∗2=96

chetan2u
nick13
How many positive integer divisors does 8! have?

a. 14
b. 15
c. 24
d. 56
e. 95

Positive integers mean 1 and above. So the smallest is 1 and largest is 8!.

Now, the method to find positive integer divisors is as shown by gmatophobia
Get the number in its prime factors and take product of all exponents after adding one to each exponent.

Number of 2s: \(\frac{8}{2}+\frac{8}{2^2}+\frac{8}{2^3}\)=4+2+1=7
Number of 3s: \(\frac{8}{3}+\frac{8}{3^2}\)=2+0=2
Number of 5s and 7s will be 1 each.

Thus, 8!=\(2^7*3^2*5*7\) and factors are \((7+1)(2+1)(1+1)(1+1)=8*3*2*2=96\)

Clearly, Option E has been wrongly written as 95. Moreover, an odd number of positive factors means the number is Perfect Square and 8! is clearly not a perfect square.
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How many positive integer divisors does 8! have? 07 Dec 2025, 06:15
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