Problem 105 - Intermediate Algebra

Question

3 because 8 is a perfect cube. $$ 54 $$ # Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as 8 # 3 because 8 is a perfect cube. $$ 54 $$

Step-by-Step Solution

Answer

Write 54 as 27 # 2, where 27 is a perfect cube.

Step 1: Understand the Concept

To express a number as a product of two integers where one is a perfect cube, identify perfect cubes that divide the number. Perfect cubes are numbers like \(1, 8, 27, 64, \ldots\).

Step 2: Identify Perfect Cubes Less Than 54

The perfect cubes less than 54 are 1, 8, and 27. Among these, we need to determine which divides 54.

Step 3: Check Each Perfect Cube

First, check if 54 is divisible by 8. Since 54 divided by 8 does not yield an integer, 8 is not a factor of 54. Next, check if 54 is divisible by 27. 54 divided by 27 equals 2, which is an integer.

Step 4: Write 54 as a Product

Since 54 is divisible by 27, write 54 as the product of 27 and 2, i.e., \(54 = 27 \times 2\). Here, 27 is a perfect cube.

Key Concepts

Integer Factorization

Integer factorization is the process of breaking down a whole number into a product of smaller integers, known as factors. When dealing with factorization, it's crucial to understand that the integer being factored can be divided exactly by its factors without leaving a remainder. Each factorization must satisfy this crucial condition.

To start with integer factorization, you simply need to:

• Identify the smallest prime number that can divide the integer.
• Perform the division and continue factoring the quotient until only prime numbers remain as factors.

This might sound simple, yet factorization is a foundational concept used across various complex mathematical and real-world applications, including cryptography and problem-solving techniques.

For example, to factorize 54, we identify that 2 is the smallest prime that divides it, yielding 27. Then, we continue with 27 and get it divided by 3, which is another prime number, leaving us with 9, which can also be divided by 3 twice. This gives us the factorization: \[54 = 2 \times 3 \times 3 \times 3\].

Divisibility Rules

Divisibility rules help us determine if one number is divisible by another without having to perform the full division. These rules are shortcuts that we can use to quickly check whether an integer can be divided evenly by another integer. Utilizing divisibility rules is particularly helpful in identifying factors of larger numbers.

Here are some simple divisibility rules that come in handy:

• A number is divisible by 2 if it is even.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 5 if its last digit is 0 or 5.
• A number is divisible by 9 if the sum of its digits is divisible by 9.

These rules can greatly speed up the process of identifying divisors, which is especially useful when working with perfect cubes, as seen in the original problem. Instead of manually testing each division by calculation, apply these rules to efficiently filter potential factors from a given set of numbers.

Perfect Cube Identification

Perfect cubes are special numbers that result from multiplying an integer by itself twice. A significant part of identifying a perfect cube involves understanding that these numbers have a specific cube root which is an integer. Some examples of perfect cubes include \(1^3 = 1, 2^3 = 8, 3^3 = 27,\) and so on.

To identify if a number is a perfect cube, you can:

• Find the cube root of the number and check if it's an integer.
• Check through known perfect cubes to see if the number matches any.

In practice, for a number like 54, you would compare it against perfect cubes less than itself, such as 1, 8, and 27, to determine potential divisors. If a number like 54 is decomposed into a product where one factor is a perfect cube (27 times 2), we see clearly how this aspect is important for solving such exercises. Identifying and applying perfect cube properties simplifies forming the product solution needed.