The number of factors of 72 is 

The process of finding the number of factors of 72 involves two main steps: prime factorization and using a formula to calculate the total factors.

Step 1: Prime Factorization of 72

To determine the prime factors of 72, we repeatedly divide the number by its smallest prime divisors:

• Divide 72 by 2 (the smallest prime number): 72 ÷ 2 = 36
• Divide 36 by 2: 36 ÷ 2 = 18
• Divide 18 by 2: 18 ÷ 2 = 9
• 9 is not divisible by 2, so we move to the next prime number, 3.
• Divide 9 by 3: 9 ÷ 3 = 3
• Finally, divide 3 by 3: 3 ÷ 3 = 1

Thus, the prime factorization of 72 is:

72 = 2³ × 3²

Step 2: Formula for the Number of Factors

The formula to calculate the total number of factors of a number from its prime factorization is:

(a + 1)(b + 1)..., where a, b, etc., are the powers of the prime factors.

For the prime factorization of 72:

• The power of 2 is 3 (from 23), so a = 3.
• The power of 3 is 2 (from 32), so b = 2.

Step 3: Calculate the Number of Factors of 72

Using the formula, substitute the values:

Number of factors = (3 + 1)(2 + 1)

Perform the calculations:

Number of factors = 4 × 3 = 12

Conclusion

The number of factors of 72 is 12. These factors are derived using the prime factorization and formula described above. Understanding the factors of 72 helps in various mathematical problems and applications. If you break down the factors of 72, they include 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.