Question

Write 60 using factorials?

Answer 1

Hint: We will first find the prime factorization of 60 and then try to find numbers in the form of 2, 3 and then 4 and so on so that we form some factorial and thus have the answer.

Complete step by step answer:
We are given that we need to write 60 as factorials.
Let us find the prime factorization of 60 as follows:-
We can write 60 as: $60 = 2 \times 2 \times 3 \times 5$
Now, we have two 2’s, one 3 and one 5 in the prime factorization of 60.
Now, since we know that: n! = 1 . 2 . 3 ……. (n – 1)
Now, we have 3, 4 and 5 in the prime factorization of 60 because two times two is 4.
Now, we can write: $60 = 3 \times 4 \times 5$
Now, we will multiply and divide the right hand side of the above equation by 2 to obtain the following equation:-
$ \Rightarrow 60 = \dfrac{{2 \times 3 \times 4 \times 5}}{2}$
Now, we can also write this as:-
$ \Rightarrow 60 = \dfrac{{1 \times 2 \times 3 \times 4 \times 5}}{2}$
Using the definition of factorial as mentioned above which is n! = 1 . 2 . 3 ……. (n – 1), we will obtain the following expression:-
$ \Rightarrow 60 = \dfrac{{5!}}{2}$

Hence, 60 is half of the 5!.

Note: Remember, n! = 1 . 2 . 3 ……. (n – 1)
We did not reach to any integral factorial but half of it, if it would have been 120, we would not have tackled any problem but to obtain a factorial of 5, one 2 was missing and to get a factorial of 4, one 2 was lying extra in there, so we somehow did something with them to manage in between.
Note that in the last second step, we did multiply the numerator by 1 without making any change because 1 is the multiplicative identity in real numbers. Multiplying any number by it or dividing any number by it does not create any difference.