Question

The number $(36)!$ is divisible by
A.${2^{36}}$
B.${2^{38}}$
C.${(12)^{17}}$
D.${(24)^{36}}$

Answer 1

Hint: We will find the value for exponents of 2 and find the exponent of 3 using some rules for this question and we will be multiplying two values and applying some rules to find the answer. Question has a factorial value so here using a factorial formula.

Complete step-by-step answer:
Here the question is
$ = (36)!$
Factorial value is written as ${2^n}$ so we will take Exponent of $2$ in $36!$ is

$ = \left[ {\dfrac{{36}}{2}} \right] + \left[ {\dfrac{{36}}{4}} \right] + \left[ {\dfrac{{36}}{8}} \right] + \left[ {\dfrac{{36}}{{16}}} \right] + \left[ {\dfrac{{36}}{{32}}} \right]$
In case we have dfractional number, we will be changed into whole number.
$ = 18 + 9 + 4 + 2 + 1 = 34$[Here $\dfrac{{36}}{8}$is $4.5$ we will take whole number $4$, $\dfrac{{36}}{{16}}$ is $2.25$ we will take whole number $2$ .$\dfrac{{36}}{{32}}$is $1.125$ we will take whole number $1$]
$34$ is changing into ${(2)^{34}}$-------$(I)$
Here factorial value written as ${3^n}$ So we will take Exponent of $3$ in $36!$ is

$ = \left[ {\dfrac{{36}}{3}} \right] + \left[ {\dfrac{{36}}{9}} \right] + \left[ {\dfrac{{36}}{{27}}} \right]$
$ = 12 + 4 + 1 = 17$
$17$ is changed into ${(3)^{17}}$ -------$(II)$
We will substitute answers in $(I),(II)$
${(2)^{34}}$.${(3)^{17}}$
Here ${(2)^{34}}$ changing into ${(2)^{17}}.{(2)^{17}}.{(3)^{17}}$
$ = {(4)^{17}}.{(3)^{17}}$
$36!$ is divisible by ${(12)^{17}}$
Here the answer is option is $c$.

Additional Information:
Factorial, in mathematics, the product of all positive integers less than or equal to a given positive integer and denoted by that integer and an exclamation point. Thus, factorial seven is written $7!$ .
The factorial of $0$ is $1$, or in symbols, \[0!{\text{ }} = {\text{ }}1\].

Note: We take the product of two exponentials with the same base, we simply add the exponents. If we take the quotient of two exponentials with the same base, we simply subtract the exponents. We can raise exponentially to another power, or take a power of a power. The result is a single exponential where the power is the product of the original exponents.