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Number Theory Level 3

Find the number of positive divisors of $10!$ .

Notation : $!$ denotes the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .

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The answer is 270.

2 solutions

Swapnil Das
Jun 14, 2016

$10!=1\times 2\times 3\times 4\times 5\times 6\times 7\times 8\times 9\times 10={ 2 }^{ 8 }\times { 3 }^{ 4 }\times { 5 }^{ 2 }\times { 7 }^{ 1 }$ .

$\therefore$ $\text{ Number of divisors}$ $= 9\times 5\times 3\times 2=270$ .

Relevant article

this problem made made me write a piece of code for this program in c

Shashank Rustagi - 5 years ago

Ahahah. I've just done the same, but in pascal😂

Alex Misiurina - 4 years, 12 months ago

Follow up: How many primes are contained in 10! ?

Mehul Arora - 4 years, 12 months ago

obviously 4 [2,3,5,7]

fahim saikat - 4 years, 12 months ago

What is that supposed to mean? I mean , I don't know the syntax of the operator you mentioned

Mehul Arora - 4 years, 12 months ago

Refb Pe
Jun 15, 2016

The prime divisor numbers of n! equals to [n/p]+[n/p^2]+[n/p^3]...... The prime numbers less than 10 are 2,3,5and7. 2: 5+2+1+0+0+........=8 3:3+1=4 5: 2 7: 1 All the divisors are the combine of these primes. Use the fomula: number=(8+1)(4+1)(2+1)(1+1)=270

Where this formula came from?

Daniel Brazil - 4 years, 12 months ago

You mean the number of divisors after knowing the number of each prime divisor?

Refb Pe - 4 years, 12 months ago

You can visualise the formula like this: Suppose we have $360 = 2^{3}.3^{2}.5^{1}$

While counting the number of divisors, we have 4 options to choose power of 2 (viz 0, 1, 2 or 3)

We have 3 options to choose power of 3 (viz 0, 1 or 2)

We have 2 options to choose power of 5 (0 or 1)

So in total we have 4 x 3 x 2 = 24 divisors to choose from (this will include 1 when we choose all bases with 0 power; and 360 when we choose all bases with their full powers)

In general, $N = 2^{a}.3^{b}.5^{c}.......$ will have (a+1)(b+1)(c+1)...... divisors, including 1 and N itself.

Rohit Sachdeva - 4 years, 12 months ago

Burn your Calculator!

Do you like challenging your brain? Enter the world where Calculators will not help

The answer is 270.

2 solutions

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