Prime and Composite Numbers

We know all prime numbers except 2 are odd.

If we multiply 50 prime numbers except 2, the product will be odd.

But if one of the prime numbers is 2, the product will be even.

Hence, 3.

The only even prime number is 2.

If we calculate the sum of 51 distinct prime numbers except 2, the sum will always be odd.

Sum of 51 prime numbers can be even only if one of the primes numbers is definitely 2.

Hence, 2.

If all the composite numbers are even, the sum will be even.

If all the composite numbers are odd, the sum will be odd.

Hence, 3.

There are 25 prime number from 1 till 100.

[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

Hence, 25.

Prime numbers are those numbers which have exactly 2 factors i.e., 1 and the number itself.

Option (a): 391 = 17 × 23. Factors are 1, 17, 23 and 391. Hence not a prime number.

Option (b): 299 = 13 × 23. Factors are 1, 13, 23 and 299. Hence not a prime number.

Option (c): 1001 = 7 × 11 × 13. Factors are 1, 7, 11, 13 etc.. Hence not a prime number.

Option (d): 997 is a prime number.

Hence, option (d).

Co-prime numbers are those pair of numbers which have no common factor apart from 1.

Option (a): 12 and 99 have 3 as common factor, hence they are not co-prime

Option (b): 36 and 81 have 9 as common factor, hence they are not co-prime

Option (c): 25 and 36 have no common factor apart from 1, hence they are co-prime

Hence, option (c).

Givne, p + 1 is a perfect square.

Let p + 1 = n²

⇒ p = n² - 1

⇒ p = (n + 1)(n - 1)

Now, if p is a prime number, the only wasy of writing p as a product of two numbers is p × 1

∴ (n + 1) = p and (n - 1) = 1

⇒ n = 2 and p = 3

∴ Only one value of p is possible.

Hence, 1.