Compute the number of positive integer divisors of 10

  • Thread starter Thread starter RM86Z
  • Start date Start date
  • Tags Tags
    Integer Positive
AI Thread Summary
The discussion focuses on calculating the number of positive integer divisors of 10! using its prime factorization. The correct factorization of 10! is identified as 2^8 x 3^4 x 5^2 x 7, leading to the calculation of divisors based on the formula for counting divisors. Initially, a mistake was made in counting the factors of 7, but it was later corrected to include it as two, resulting in the correct total of 270 divisors. The confusion also arose from the mention of 270! being greater than 10!, which was clarified as irrelevant to the divisor count. The final answer for the number of positive integer divisors of 10! is confirmed to be 270.
RM86Z
Messages
23
Reaction score
6
Homework Statement
number of positive integer divisors of 10!
Relevant Equations
10!
Compute the number of positive integer divisors of 10!. By the fundamental theorem of arithmetic and the factorial expansion:

10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
= 2 x 5 x 3^2 x 7 x 2 x 3 x 5 x 2^2 x 3 x 2 x 1
= 2^8 x 3^4 x 5^2 x 7

Then there are 9 possibilities for 2, 5 for 3, 3 for 5 and for 7 giving 9 x 5 x 3 = 135.

The book gives 270 as the answer, where am I going wrong?

Thank you!

EDIT:Oops, I should have counted 7 as two giving 9 x 5 x 3 x 2 = 270!
 
Reply
  • Like
Likes .Scott and Delta2
Physics news on Phys.org
RM86Z said:
Homework Statement:: number of positive integer divisors of 10!
Relevant Equations:: 10!

270!
Isn't that greater than 10! ?
 
Reply
  • Haha
  • Wow
Likes SammyS, RM86Z and Delta2

Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
View full post »

Thread 'Making the denominator of a certain fraction real'

Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
View full post »

Similar threads

Find integer points on this equation
Replies
8
Views
1K
Are these values of the following correct? (Complex cube root of unity)
Replies
9
Views
2K
Proof That an Integer is Divisible by 2
Replies
3
Views
2K
Obtain the digits ## x ## and ## y ##
Replies
3
Views
1K
Divisibility of Integers ## 176521221 ## & ## 149235678 ## by 9 & 11
Replies
9
Views
2K
For any integer ## a ##, the units digit of ## a^{2} ## is?
Replies
7
Views
2K
Any one of the integers ## 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ## can occur?
Replies
3
Views
1K
Finding Integer with Chinese Remainder Theorem
Replies
3
Views
2K
Find an integer having the remainders ## 1, 2, 5, 5 ##
Replies
7
Views
2K
Show that ## a^{2}-a+7 ## ends in one of the digits ## 3, 7 ,9##
Replies
6
Views
2K

Hot Threads

Recent Insights

Back
Top