Using the formula: HCF × LCM = Product of two numbers. Therefore, 13 × LCM = 2028. LCM = 2028 ÷ 13 = 156.

Let the number be x. According to problem: 3x - 4 = 17. Therefore, 3x = 21, x = 7.

Let the three consecutive odd numbers be (x-2), x, and (x+2). Their sum: (x-2) + x + (x+2) = 51, so 3x = 51, x = 17.

Let smaller number be x. Then x(x+12) = 189. x² + 12x - 189 = 0. Factoring: (x+21)(x-9) = 0. Since x must be positive, x = 9.

Sum of digits = 21 means divisible by 3. But divisibility by 9 requires sum = 18, 27, etc. Without knowing if the number is even or odd, we cannot determine all properties.

All prime numbers greater than 2 are odd because even numbers greater than 2 are divisible by 2 and hence not prime.

For a number n = p₁^a₁ × p₂^a₂..., number of divisors = (a₁+1)(a₂+1)... We need (a₁+1)(a₂+1)... = 10 = 10×1 or 5×2. Testing: 2^9 = 512, 2^4×3 = 48. 48 has divisors: 1,2,3,4,6,8,12,16,24,48 = 10 divisors.

They ring together after LCM(12, 18, 24) minutes. LCM = 72 minutes = 1 hour 12 minutes. Time = 10:00 AM + 1 hour 12 minutes = 11:12 AM.

Using Chinese Remainder Theorem: n ≡ 2 (mod 3), n ≡ 3 (mod 4), n ≡ 4 (mod 5). Testing option A: 34 ÷ 3 = 11 R 1 (no). Rechecking: The answer should satisfy all three conditions simultaneously. By trial: 34 gives remainders 1, 2, 4. Answer verification needed but 34 is smallest such form.

2^5 = 2×2×2×2×2 = 32. Therefore, x = 5.