Factors and Factorials Level - 2

If a three-digit natural number has exactly 3 factors, how many such three-digit numbers exist?

Find the product of all the factors of N = 1440. Express your answer in the form 1440^k. What is the value of k?

How many factors of N = 2⁵ × 3⁴ × 5³ are divisible by 12 but not divisible by 24?

Find the sum of all the factors of N = 240 that are of the form 4k + 2, where k is an integer ≥ 0.

Let S be the set of all factors of N = 2³ × 3⁴ × 5². If two distinct factors are chosen at random from S, what is the probability that their product is a multiple of 10?

A locker room contains 1000 lockers numbered 1 to 1000. The 1st employee opens every locker. The 2nd employee closes every even-numbered locker. The 3rd employee changes the state of every locker that is a multiple of 3. This continues up to the 1000th employee. At the end, how many lockers remain open?

Find the smallest natural number that has exactly 18 factors.

Find the smallest natural number that has exactly 15 factors and is a multiple of 6.

How many factors of N = 6⁶ × 15⁵ × 10⁴ are perfect squares?

How many factors of N = 2⁹ × 3⁷ × 5⁵ × 7³ are perfect cubes?