Calculating Number of Factors

When referring to the factors of a number, we typically mean the positive factors. How do we calculate the number of factors a specific number has?

• As usual, we start by finding the prime factorization of the number
• We then list the exponent of each prime factor and add 1 to each exponent
• Finally, we multiply each sum together

Example: How many factors does the number 90 have?

• 90 = 2¹ * 3² * 5¹
• The exponent of each prime factor is 1, 2, and 1. Adding one to each we get (1 + 1), (2 + 1), and (1 + 1)
• Multiplying the sums together we get (1 + 1) * (2 + 1) * (1 + 1) = (2) * (3) * (2) = 12
• 90 has 12 positive factors

Example: How many factors does the number 1932 have?

• 1932 = 2² * 3¹ * 7¹ * 23¹
• (2 + 1) * (1 + 1) * (1 + 1) * (1 + 1) = (3) * (2) * (2) * (2) = 24
• 1932 has 24 positive factors

* Example: What is the smallest positive integer that has 10 factors?

• We can observe that 10 = 2 * 5, meaning that the number in question has just two prime factors in its decomposition – one with the exponent of α = 1, the other of β = 4: N = (p¹)(q⁴). To make N as small as possible, we have to choose the smallest available primes, 2 and 3. The answer then must be N = (3¹)(2⁴) = 48
• The smallest positive integer that has 10 factors is the number 48

* Source: https://thepuzzlr.com/quizzes/can-you-find-the-number-of-factors/