Which of the following numbers are prime?

(i) 29

(ii) 57

(iii) 43

(iv) 61

(i) 29

As 5 × 5 = 25 < 29 and 6 × 6 = 36 > 29, so 5 is the largest number such that 5 × 5 ≤ 29.

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

29 is not divisible by 2, 3 or 5.

Hence, 29 is a prime number.

(ii) 57

Sum of digits of 57 = 5 + 7 = 12, which is divisible by 3.

So, 57 is divisible by 3 (57 = 3 × 19).

Hence, 57 is not a prime number.

(iii) 43

As 6 × 6 = 36 < 43 and 7 × 7 = 49 > 43, so 6 is the largest number such that 6 × 6 ≤ 43.

The prime numbers less than or equal to 6 are 2, 3 and 5.

43 is not divisible by 2, 3 or 5.

Hence, 43 is a prime number.

(iv) 61

As 7 × 7 = 49 < 61 and 8 × 8 = 64 > 61, so 7 is the largest number such that 7 × 7 ≤ 61.

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

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

Hence, 61 is a prime number.