How many factors does 180 have?

Steps to Solve

1. Find the Prime Factorization

To find the prime factorization of 180, we can divide it by the smallest prime number and continue dividing the quotient until we reach 1.

Start with 180: $$ 180 \div 2 = 90 \ 90 \div 2 = 45 \ 45 \div 3 = 15 \ 15 \div 3 = 5 \ 5 \div 5 = 1 $$ Thus, the prime factorization is: $$ 180 = 2^2 \times 3^2 \times 5^1 $$

2. Use the Prime Factorization to Find the Number of Factors

To find the number of factors, you use the formula based on the prime factorization. If a number is represented as: $$ n = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k} $$ then the number of factors of ( n ) is given by: $$ (e_1 + 1)(e_2 + 1)...(e_k + 1) $$

Now apply it to our prime factorization:

• For (2^2), ( e_1 = 2 ) so ( e_1 + 1 = 3 )
• For (3^2), ( e_2 = 2 ) so ( e_2 + 1 = 3 )
• For (5^1), ( e_3 = 1 ) so ( e_3 + 1 = 2 )

Hence, the total number of factors is: $$ (2+1)(2+1)(1+1) = 3 \times 3 \times 2 $$

3. Calculate the Total Number of Factors

Now, let’s calculate it: $$ 3 \times 3 \times 2 = 18 $$