Plz someone help me to solve this problem.

[Mathlover]

How many different positive integers divide 10! ?

[Mohaimin]

I am not giving the complete solution. Just telling one possible approach. Do the solution yourself or ask if you need more information.

First note that 10! has only the prime factors 2, 3, 5 and 7. The next prime 11 cannot be in 10! because it is bigger than 10.
Now you need to find out the maximum power of each of the primes that divides 10!.
So, you get the Prime Power Factorisation(PPF) of 10!
There is a way to find out number of factors from PPF.

[Mathlover]

the question was to find positive integers,not prime integers.

[Nadim Ul Abrar]

You need to know the theorem that,
If n is any number such that
n=p^{a}p'^{b}........
Where p,p'.......
Then
the number of factors will be
(a+1)(b+1).....
And thats the number of divisor tooo;)

[Nadim Ul Abrar]

The ans will be

270

[Mathlover]

thanks nadim......

[Nadim Ul Abrar]

Thanks to u too . Cause u'r the first one thanked in this forum

[Tahmid Hasan]

tao function is very helpful here i think.

[Mohaimin]

the question was to find positive integers,not prime integers.

I know, I did not give you the complete solution. Just gave some hints. I meant to solve it exactly how Nadim solved.

You need to know the theorem that,
If n is any number such that
n=p^{a}p'^{b}........
Where p,p'.......
Then
the number of factors will be
(a+1)(b+1).....
And thats the number of divisor tooo;)

p, p`... should be prime. You did not mention that.

[Nadim Ul Abrar]

Sorry @ MOhaimin