To find the number by which 4608 should be divided to get a perfect cube, we must factorize 4608 and then adjust the factors so that each prime factor appears in multiples of 3, since a perfect cube has every prime factor raised to an exponent that is a multiple of 3.

Step 1: Prime factorization of 4608

Let's factorize 4608:

$4608 \div 2 = 2304$ $2304 \div 2 = 1152$ $1152 \div 2 = 576$ $576 \div 2 = 288$ $288 \div 2 = 144$ $144 \div 2 = 72$ $72 \div 2 = 36$ $36 \div 2 = 18$ $18 \div 2 = 9$ $9 \div 3 = 3$ $3 \div 3 = 1$

So, the prime factorization of 4608 is:

$4608 = 2^8 \times 3^2$

Step 2: Adjusting for a perfect cube

To make 4608 a perfect cube, the exponents of the prime factors must be multiples of 3.

• The exponent of 2 is 8, which is not a multiple of 3. To make it a multiple of 3, we need to divide 4608 by $2^2 = 4$ (since $8 - 6 = 2$).
• The exponent of 3 is 2, which is not a multiple of 3. To make it a multiple of 3, we need to divide 4608 by $3^2 = 9$.

Step 3: Final division

Now, we need to divide 4608 by the product of $2^2$ and $3^2$:

$2^2 \times 3^2 = 4 \times 9 = 36$

Thus, 4608 should be divided by 36 to get a perfect cube.

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Related Questions:

1. What is a perfect cube and how is it defined?
2. Can you explain prime factorization in more detail?
3. How do you verify if a number is a perfect cube?
4. What are other examples of numbers that need adjustments to become perfect cubes?
5. How would you factorize larger numbers efficiently?

Tip:

A perfect cube is a number that can be expressed as $n^3$, where $n$ is an integer. Each prime factor's exponent must be divisible by 3.