Find HCF of 48 and 64 using Euclidean algorithm: 64 = 48 × 1 + 16; 48 = 16 × 3 + 0. Therefore, HCF = 16.

12 = 2² × 3; 18 = 2 × 3². LCM = 2² × 3² = 4 × 9 = 36.

61 is only divisible by 1 and itself. 51 = 3 × 17, 57 = 3 × 19, 63 = 9 × 7 are composite numbers.

72 = 2³ × 3². Number of divisors = (3+1)(2+1) = 4 × 3 = 12.

Since 9 and 11 are coprime (HCF = 1), the number must be divisible by 9 × 11 = 99.

Let numbers be x, x+2, x+4, x+6, x+8. Sum = 5x + 20 = 120, so x = 20. Largest = 20 + 8 = 28.

Numbers divisible by both 6 and 9 must be divisible by LCM(6,9) = 18. Between 1-100: 18, 36, 54, 72, 90. Count = 5.

Digit sum = 9 + 9 + 9 + 9 = 36. Note: A number and its digit sum have the same remainder when divided by 9.

Let numbers be a and b. (a+b)² - 4ab = (a-b)². So (a-b)² = 400 - 384 = 16, thus |a-b| = 4.

Using Legendre's formula: ⌊27/3⌋ + ⌊27/9⌋ + ⌊27/27⌋ = 9 + 3 + 1 = 13.