Question

Use prime factorisation to determine the smallest value that 55125 must be multiplied by to give a cube number.

Answer 1

The smallest value that 55125 must be multiplied by to give a cube number is 7.

Step-by-step explanation:

We need to find the prime factorization of 55125.

Prime factorization of 55125:

55125 = 3 * 5^3 * 7^2

For a number to be a perfect cube, all the prime factors must have an exponent that is a multiple of 3.

Exponent of 3 in the prime factorization: 5^3 = 125 (which is already a perfect cube)

Exponent of 7 in the prime factorization: 7^2 = 49 (not a multiple of 3)

To make the exponent of 7 a multiple of 3, we need to multiply 55125 by 7.

Therefore, the smallest value that 55125 must be multiplied by to give a cube number is 7.

Answer 2

To determine the smallest value that 55125 must be multiplied by to give a cube number using prime factorization, we find the prime factorization of 55125, look at the exponents of the prime factors, and multiply the number by 3 × 3 × 3 × 11 × 67.

Step-by-step explanation:

The first step in using prime factorization to determine the smallest value that 55125 must be multiplied by to give a cube number is to find the prime factorization of 55125. We start by dividing 55125 by the smallest prime number, which is 2. Since 55125 is not divisible by 2, we move on to the next prime number, which is 3. 55125 ÷ 3 = 18375. We continue dividing by prime numbers until we reach a quotient of 1. The prime factorization of 55125 is 3 × 5 × 5 × 11 × 67. Now we need to find the smallest value that must be multiplied by 55125 to give a cube number. We look at the exponents of the prime factors in the factorization: 3 (from 3), 2 (from 5 × 5), 1 (from 11), and 1 (from 67). To make all the exponents divisible by 3, we need to multiply the number by 3 × 3 × 3 × 11 × 67 = 2079. Therefore, the smallest value that 55125 must be multiplied by to give a cube number is 2079.