Divisor Calculation for 120

To find the number of divisors of 120, we first find its prime factorization.

Prime Factorization of 120

120 can be broken down into its prime factors:

• $120 = 12 \times 10$
• $12 = 2 \times 6 = 2 \times 2 \times 3 = 2^2 \times 3^1$
• $10 = 2 \times 5 = 2^1 \times 5^1$
• Combining these, $120 = (2^2 \times 3^1) \times (2^1 \times 5^1) = 2^{2+1} \times 3^1 \times 5^1 = 2^3 \times 3^1 \times 5^1$.

The prime factorization of 120 is $2^3 \times 3^1 \times 5^1$.

Number of Divisors Formula

If the prime factorization of a number $N$ is $p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}$, the total number of divisors is given by the product of one more than each exponent:

Number of Divisors = $(a_1+1)(a_2+1)\dots(a_k+1)$

Calculating the Number of Divisors

For 120, the prime factorization is $2^3 \times 3^1 \times 5^1$. The exponents are 3, 1, and 1.

• Using the formula, the number of divisors is $(3+1) \times (1+1) \times (1+1)$.
• Calculation: $4 \times 2 \times 2 = 16$.

Therefore, the number of divisors of 120 is 16.