Question

The number of divisors of \[7!\] is
A) 24
B) 72
C) 64
D) 60

Answer 1

Hint:
Divisors mean the number that divides the given number completely keeping remainder equal to zero. First we will find the value of \[7!\] and then we will write the product in the powers of prime numbers. And adding one to each power and then taking their product will give us the number of divisors.

Complete step by step solution:
We know that if we have a number N given in the form N! and we are asked to find the divisors of N! Then ,
First we will write the number in the product of prime numbers.
\[N = a \times b \times c\] where a, b, c are the prime numbers.
So
\[
  7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \\
   \Rightarrow 7 \times 2 \times 3 \times 5 \times 2 \times 2 \times 3 \times 2 \\
\]

Then we will write the product in the form of a product of prime numbers with powers.
\[ \Rightarrow {a^x}{b^y}{c^z}\]
So,
\[ \Rightarrow {2^4} \times {3^2} \times {5^1} \times {7^1}\]

At last we will add 1 to each power and will take the product of them and it will give us the number of divisors of N!
\[ \Rightarrow \left( {x + 1} \right)\left( {y + 1} \right)\left( {z + 1} \right)\]
So,
\[ \Rightarrow \left( {4 + 1} \right)\left( {2 + 1} \right)\left( {1 + 1} \right)\left( {1 + 1} \right)\]
And now we will take the product of it.
\[
   \Rightarrow 5 \times 3 \times 2 \times 2 \\
   \Rightarrow 60 \\
\]
And this is the number of divisors of 7!

So the correct option is D.

Note:
In these types of problems don’t try to divide the value of factorial by separate numbers that is too time consuming and tedious work. This method mentioned above is the correct approach. Do write the product of prime numbers correctly in their powers form.