Question

The exponent of 3 in the prime factorisation of 864 is :

• A. 2
• B. 5
• C. 4
• D. 3

Hint:

To find the exponent of a prime in a number's prime factorization: 1. Divide the number repeatedly by that prime until it's no longer divisible. Alternatively, find the full prime factorization. For 864: \(864 = 2 \times 432\) \( = 2 \times 2 \times 216\) \( = 2 \times 2 \times 2 \times 108\) \( = 2 \times 2 \times 2 \times 2 \times 54\) \( = 2 \times 2 \times 2 \times 2 \times 2 \times 27\) \( = 2^5 \times 27\) Now factorize 27: \(27 = 3 \times 9 = 3 \times 3 \times 3 = 3^3\). So, \(864 = 2^5 \times 3^3\). The exponent of 3 is 3.

Answer 1

The Correct Option is D

Concept: Prime factorization is the process of finding which prime numbers multiply together to make the original number. The exponent of a prime factor is the number of times that prime factor appears in the factorization. Step 1: Perform prime factorization of 864 Start by dividing 864 by the smallest prime numbers. \[ 864 \div 2 = 432 \] \[ 432 \div 2 = 216 \] \[ 216 \div 2 = 108 \] \[ 108 \div 2 = 54 \] \[ 54 \div 2 = 27 \] Now 27 is not divisible by 2. Try the next prime number, 3. \[ 27 \div 3 = 9 \] \[ 9 \div 3 = 3 \] \[ 3 \div 3 = 1 \] So, the prime factorization of 864 is \(2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3\). Step 2: Write the prime factorization in exponential form \[ 864 = 2^5 \times 3^3 \] Step 3: Identify the exponent of 3 In the prime factorization \(2^5 \times 3^3\), the prime factor 3 has an exponent of 3. Therefore, the exponent of 3 in the prime factorization of 864 is 3.