Question

Natural number N has exactly 50 factors (including 1 and itself). To what power should the number N be raised to get the product of all of its 50 factors?

Answer 1

To find the power to which a natural number N should be raised to get the product of all its factors, we need to determine the prime factorization of N, count the powers of each prime factor, and subtract 1 from each power.

Step-by-step explanation:

To find the power to which a natural number N should be raised to get the product of all its factors, we need to determine the prime factorization of N and count the powers of each prime factor.

Let's assume the prime factorization of N is N = p1^a1 * p2^a2 * p3^a3 * ... * pn^an, where p1, p2, p3... pn are prime numbers and a1, a2, a3... an are their respective powers.

Since N has exactly 50 factors, the formula for the number of factors of N is: (a1 + 1) * (a2 + 1) * (a3 + 1) * ... * (an + 1) = 50.

Then, the power to which N should be raised to get the product of its factors is: a1 * a2 * a3 * ... * an = (50 - 1).