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: \[ \boxed{32}. \]

Answer 2

To find the number of positive divisors of 1080:

Step 1: Perform prime factorization of 1080.
\[ 1080 = 2 \times 540 = 2^1 \times 540 \]
\[ 540 = 2 \times 270 = 2^2 \times 270 \]
\[ 270 = 2 \times 135 = 2^3 \times 135 \]
\[ 135 = 3 \times 45 = 3^1 \times 45 \]
\[ 45 = 3 \times 15 = 3^2 \times 15 \]
\[ 15 = 3 \times 5 = 3^3 \times 5^1 \]

So, the prime factorization is:
\[ 1080 = 2^3 \times 3^3 \times 5^1 \]

Step 2: Use the formula for the number of positive divisors:
If
\[ n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} \]
then number of positive divisors
\[ = (a_1 + 1)(a_2 + 1) \cdots (a_k + 1) \]

Step 3: Apply to 1080:
\[ (3 + 1)(3 + 1)(1 + 1) = 4 \times 4 \times 2 = 32 \]

Therefore, the number of positive divisors of 1080 is:
\[ \boxed{32} \]