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.

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Step 1: Identify all prime numbers between 10 and 30

We need to find every number in the range (10, 30) that is only divisible by 1 and itself. Check each number: 11 (prime), 12 (divisible by 2,3,4,6), 13 (prime), 14 (divisible by 2,7), 15 (divisible by 3,5), 16 (divisible by 2,4,8), 17 (prime), 18 (divisible by 2,3,6,9), 19 (prime), 20 (divisible by 2,4,5,10), 21 (divisible by 3,7), 22 (divisible by 2,11), 23 (prime), 24 (divisible by 2,3,4,6,8,12), 25 (divisible by 5), 26 (divisible by 2,13), 27 (divisible by 3,9), 28 (divisible by 2,4,7,14), 29 (prime).

Step 2: Add all identified prime numbers

Sum the six prime numbers we identified by adding them sequentially.

The sum of all prime numbers between 10 and 30 is 112.

The answer is (B) 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