51 = 3 × 17 (not prime), 53 is prime (only divisible by 1 and 53), 55 = 5 × 11 (not prime), 57 = 3 × 19 (not prime).

Therefore, 53 is the smallest prime greater than 50.

36 = 2² × 3².

Number of factors = (2+1)(2+1) = 3 × 3 = 9.

The factors are: 1, 2, 3, 4, 6, 9, 12, 18, 36.

12 = 2² × 3, 18 = 2 × 3², 24 = 2³ × 3. LCM = 2³ × 3² = 8 × 9 = 72.

∛512 = ∛(8³) = 8, since 8 × 8 × 8 = 512.

Odd numbers between 10 and 50: 11, 13, 15, ..., 49.

This is an AP with first term 11, last term 49, and common difference 2.

Number of terms = (49-11)/2 + 1 = 19 + 1 = 20.

LCM(9, 11) = 99 (since 9 and 11 are coprime).

Smallest three-digit multiple of 99 is 99 × 2 = 198.

20 = 2² × 5.

Sum of divisors = (1 + 2 + 4)(1 + 5) = 7 × 6 = 42.

Divisors are: 1, 2, 4, 5, 10, 20.

32 ÷ 7 = 4 remainder 4 (No). 33 ÷ 7 = 4 remainder 5 (No). 34 ÷ 7 = 4 remainder 6 (No). 32 = 7×4 + 5 = 28 + 5 (Yes). 32 ÷ 7 gives remainder 5.

Digital root = 9 + 8 + 7 + 6 = 30 → 3 + 0 = 3.

Alternatively, since the sum is divisible by 9, the digital root is 9.

Wait: 9+8+7+6 = 30, 3+0 = 3.

So digital root is 3.

Using HCF × LCM = Product of two numbers: 15 × 180 = 45 × x.

Therefore, 2700 = 45x, so x = 60.