Which of the following numbers are prime:

(i) 101

(ii) 251

(iii) 323

(iv) 397

(i) 101

As 10 × 10 = 100 < 101 and 11 × 11 = 121 > 101, so 10 is the largest number such that 10 × 10 ≤ 101.

The prime numbers less than or equal to 10 are 2, 3, 5 and 7.

101 is not divisible by 2, 3, 5 or 7.

Hence, 101 is a prime number.

(ii) 251

As 15 × 15 = 225 < 251 and 16 × 16 = 256 > 251, so 15 is the largest number such that 15 × 15 ≤ 251.

The prime numbers less than or equal to 15 are 2, 3, 5, 7, 11 and 13.

251 is not divisible by 2, 3, 5, 7, 11 or 13.

Hence, 251 is a prime number.

(iii) 323

Note that 17 × 19 = 323, so 17 and 19 are factors of 323.

Hence, 323 is not a prime number.

(iv) 397

As 19 × 19 = 361 < 397 and 20 × 20 = 400 > 397, so 19 is the largest number such that 19 × 19 ≤ 397.

The prime numbers less than or equal to 19 are 2, 3, 5, 7, 11, 13, 17 and 19.

397 is not divisible by 2, 3, 5, 7, 11, 13, 17 or 19.

Hence, 397 is a prime number.