Using the division algorithm: Number = (Divisor × Quotient) + Remainder. Number = (15 × 23) + 8 = 345 + 8 = 353.

Using the property: GCD(a,b) × LCM(a,b) = a × b. Therefore: 12 × 144 = 36 × b. So 1728 = 36b, which gives b = 48.

Let number be 10a + b. Reversed number is 10b + a. Given: (10b + a) - (10a + b) = 27, so 9b - 9a = 27, thus b - a = 3. Also a + b = 9. Solving: b = 6, a = 3. Original number = 36.

Let three consecutive odd numbers be x, x+2, x+4. Their sum: x + (x+2) + (x+4) = 51. So 3x + 6 = 51, thus 3x = 45, and x = 15.

Prime factorizations: 24 = 2³ × 3, 36 = 2² × 3², 60 = 2² × 3 × 5. LCM = 2³ × 3² × 5 = 8 × 9 × 5 = 360.

If one number is 60 and product is 2160, then other number = 2160 ÷ 60 = 36. Sum = 60 + 36 = 96. Wait, let me verify GCD(60, 36) = 12. Yes, 12 is correct. Sum = 96.

Numbers divisible by both 4 and 6 are divisible by LCM(4,6) = 12. Numbers from 1 to 500 divisible by 12: ⌊500/12⌋ = 41.666..., so 41 numbers. Actually, ⌊500÷12⌋ = 41, but we need to check: 12 × 41 = 492. So there are 41 numbers. Let me recalculate: 500 ÷ 12 = 41.666, so answer is 41. Wait, the options suggest 42. Let me verify: counting from 12, 24, 36...492. That's 492/12 = 41. The correct count is 41, but closest option is 42.

Factoring: x² - 5x + 6 = (x - 2)(x - 3) = 0. Therefore x = 2 or x = 3.

Let number be n. n ≡ 4 (mod 7) and n ≡ 6 (mod 11). From first: n = 7k + 4. Substituting in second: 7k + 4 ≡ 6 (mod 11), so 7k ≡ 2 (mod 11). Testing values: k = 4 gives 7(4) + 4 = 32 ≡ 10 (mod 11). Try k = 5: 7(5) + 4 = 39 ≡ 6 (mod 11). Yes, 39 works.

A number is divisible by 9 if sum of its digits is divisible by 9. Here sum is 12, which is not divisible by 9. However, any number divisible by 9 is also divisible by 3. But the given condition states sum of digits is 12, and divisible by 9, which is contradictory. Re-reading: if divisible by 9, then sum must be divisible by 9. Since sum is 12 and divisible by 3, the number is divisible by 3.