Answer:
2
Step-by-step explanation:
Definitions
- Quotient: The result obtained by dividing one number by another.
- Perfect cube: The result obtained by multiplying the same integer three times.
- Prime number: a whole number greater than 1 that cannot be made by multiplying other whole numbers.
- Prime Factorization: prime numbers that multiply together to make the original number.
Find the prime factorization of 6750
The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, ...
To find which prime numbers multiply together to make 6750, start by dividing 6750 by the first prime number, 2:
⇒ 6750 ÷ 2 = 3375
As 3375 is not a prime number, we need to divide again. 3375 is not divisible by 2, so let's try dividing it by the next prime number, 3:
⇒ 3375 ÷ 3 = 1125
Continue like this until the end result is a prime number:
⇒ 1125 ÷ 3 = 375
⇒ 375 ÷ 3 = 125
⇒ 125 ÷ 5 = 25
⇒ 25 ÷ 5 = 5
As 5 is a prime number, we can stop.
Therefore, 6750 is the product of:
⇒ 6750 = 2 × 3 × 3 × 3 × 5 × 5 × 5
As 3 and 5 appear three times, we can write this using exponents:
⇒ 6750 = 2 × 3³ × 5³
3³ and 5³ are perfect cubes. If they are multiplied together they make another perfect cube:
⇒ 3³ × 5³ = (3 × 5)³ = 15³
Therefore:
[tex]\sf \implies 6750=2 \times 15^3[/tex]
[tex]\sf \implies \dfrac{6750}{2}=15^3[/tex]
Therefore, the least number by which 6750 may be divided so that the quotient (result) is a perfect cube is 2.
Learn more about prime factorization here:
https://brainly.com/question/27804094
