Question

The number of positive divisors of 1080 is:

• A. 30
• B. 32
• C. 23
• D. 31

Hint:

To find the number of divisors of a number, first find its prime factorization, then apply the formula \( (e_1 + 1)(e_2 + 1) \dots (e_k + 1) \), where \( e_1, e_2, \dots, e_k \) are the exponents in the factorization.

Answer 1

The Correct Option is B

To find the number of positive divisors of a number, we first find its prime factorization. 
Step 1: 
The prime factorization of 1080 can be found by dividing it by the smallest prime number and continuing the process: \[ 1080 \div 2 = 540, \quad 540 \div 2 = 270, \quad 270 \div 2 = 135, \quad 135 \div 3 = 45, \quad 45 \div 3 = 15, \quad 15 \div 3 = 5. \] Finally, \( 5 \div 5 = 1 \). So, the prime factorization of 1080 is: \[ 1080 = 2^3 \times 3^3 \times 5. \] 
Step 2: 
The number of divisors of a number is given by the formula: \[ \text{Number of divisors} = (e_1 + 1)(e_2 + 1) \dots (e_k + 1), \] where \( e_1, e_2, \dots, e_k \) are the exponents in the prime factorization of the number. For \( 1080 = 2^3 \times 3^3 \times 5^1 \), the exponents are 3, 3, and 1. Therefore, the number of divisors is: \[ (3+1)(3+1)(1+1) = 4 \times 4 \times 2 = 32. \] Thus, the number of divisors of 1080 is: \[ \boxed{32}. \]