Question

2.
The least positive integer n such that n! divisible by 75 is
(a) 5
(b) 7.
(c) 10
(d) 75​

Answer 1

Answer:

5

Step-by-step explanation:

because 5 is divisible by 75 =5÷75=15

Answer 2

We need to recall the following definition of factorial.

• The factorial of a natural number $n$ is the product of the positive integers less than or equal to $n$.
• $n!=1\times2\times3\times.......(n-1)\times n$

Given:

$n$ is a positive integer.

$n!$ is divisible by $75$.

Option a) $5$

We get,

$\frac{5!}{75}=\frac{120}{75}=\frac{8}{5}$

$5!$ is not divisible by $75$.

Hence, this option is incorrect.

Option b) $7$

We get,

$\frac{7!}{75}=\frac{5040}{75}=\frac{336}{5}$

$7!$ is not divisible by $75$.

Hence, this option is incorrect.

Option c) $10$

We get,

$\frac{10!}{75}=\frac{3628800}{75}=48384$

$10!$ is divisible by $75$.

Hence, this option is correct.

Option d) $75$

We get,

$\frac{75!}{75}=74!$

$75!$ is divisible by $75$.

Hence, this option is correct.

Since $n$ is a least positive integer and $n!$ is divisible by $75$.

$10 < 75$.

Hence, the correct option is (c) $10$.