prove that a positive integer n is prime number if no prime p less than or equal to root n divides n

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Q. Prove that a positive integer

n

is prime number, if no prime

p

less than or equal to

\sqrt{n}

divides

n .

Q. Prove that a positive integer n is prime, if no prime p less than or equal to √n divides n.
From RD Sharma class 10 HOTS.

If n is a positive integer greater than 1, then PRO(n) represents the product of all the prime numbers less than or equal to n. The largest prime factor of PRO(12)+11 is:

Q. Let

n

be a positive integer such that

2^{n} 1

is a prime number. Prove that n is a prime number.

Q. According to Fermat's theorem,

n^{p} - n = M (p) = a

is a multiple of

p

, when

p

is a prime number and

n \in N

.
It is also known that the highest index of power of a prime number

p

which divides

n!

is equal to

[\frac{n}{p}] + [{\frac{n}{p}}^{2}] + [{\frac{n}{p}}^{3}] + \dots

,
where

[x] =

greatest integer less than or equal to

x

The highest power of

7

which divides

1000!

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