2,600 has how many positive divisors?

Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Back to the original question:
If p and q are prime numbers, how many divisors does the product \(p^3*q^6\) have?

According to above the number of distinct factors of \(2,600=2^3*5^2*13\) would be \((3+1)(2+1)(1+1)=24\).

For more on this check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it helps.