10³ = 1000. This is a 4-digit number and a perfect cube. 9³ = 729 (3-digit). So 1000 is the smallest 4-digit perfect cube.

A perfect number equals the sum of its proper divisors. For 28: proper divisors are 1, 2, 4, 7, 14. Sum = 1+2+4+7+14 = 28. So 28 is a perfect number.

For any two numbers: Product = HCF × LCM. Product = 12 × 144 = 1728.

Prime numbers between 10 and 30: 11, 13, 17, 19, 23, 29. Sum = 11 + 13 + 17 + 19 + 23 + 29 = 112

Let the number be x. According to problem: (8x)/2 = 64. Simplifying: 4x = 64. Therefore x = 16

Sum of first n natural numbers = n(n+1)/2. For n=50: Sum = 50(51)/2 = 2550/2 = 1275

Using division algorithm: Dividend = (Divisor × Quotient) + Remainder. Number = 7 × 12 + 5 = 84 + 5 = 89

Check from 99 backwards: 99=9×11 (not prime), 98=2×49 (not prime), 97 is only divisible by 1 and 97 (prime). Therefore 97 is the largest 2-digit prime

Let x and y be consecutive even numbers with x < y. Then y = x + 2. Given: x + y = 66, so x + (x+2) = 66, giving 2x + 2 = 66, thus 2x = 64, and x = 32

Let the number be x. According to problem: (5x)/12 = 20. Solving: 5x = 240, thus x = 48