Problem 104 - Intermediate Algebra

Question

3 because 8 is a perfect cube. $$ 56 $$ # 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. $$ 56 $$

Step-by-Step Solution

Answer

56 can be expressed as 8 # 7, where 8 is a perfect cube.

Step 1: Identify perfect cube factors

The first step is to list the perfect cubes less than or equal to the number we are given, which is 56. Perfect cubes that are relevant include 1, 8, and 27. Since 27 is too large, consider 8 and 1.

Step 2: Test divisibility

Check if 56 can be divided by any of these perfect cubes (8 and 1) without leaving a remainder. 56 divided by 8 gives 7, and 56 divided by 1 gives 56.

Step 3: Select the appropriate factors

Since 8 is a perfect cube and divides 56 evenly, we can write 56 as a product of 8 and the quotient we obtained previously, which is 7.

Step 4: Write in desired format

Express 56 as a product 8 # 7, reflecting that 8 is a perfect cube, similar to the example given with 24.

Key Concepts

Integer Factorization

Integer factorization is the process of breaking down a number into a product of smaller integers. The main goal is to find two or more numbers that multiply together to give the original number.
When dealing with perfect cubes, we focus on identifying one factor that is a cube of an integer, like 8, which is 2 cubed.
In our example with the number 56, we aimed to express it as a product where one factor is a perfect cube. In this case, we expressed 56 as 8 and 7, where 8 is the perfect cube (since 8 = 2^3).

• Integer factorization helps simplify and understand mathematical problems.
• This concept is also useful for simplifying higher mathematical functions, like finding least common multiples or greatest common divisors.

Understanding how to factor integers correctly is crucial for simplifying expressions and solving more complex mathematical equations.

Divisibility Rules

Divisibility rules help us quickly determine whether one number can be divided by another without leaving a remainder.
In our case, we needed to check if the number 56 could be evenly divided by perfect cubes like 8.
By applying divisibility rules, we found that when we divided 56 by 8, the result was a whole number (7), indicating that 56 is divisible by 8.

• Divisibility by 2: A number is divisible by 2 if it is even.
• Divisibility by 3: Add the digits of the number; if the sum is divisible by 3, so is the number.
• Divisibility by cubes: Check if dividing the number by the cube results in a whole number quotient.

Utilizing these rules saves time and provides a clear path to determining factors, especially important when identifying perfect cube factors.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators that represent a value.
When expressing numbers using mathematical expressions that involve factors, we are often interested in simplifying or breaking down numbers into more manageable parts.
In our example, expressing 56 as 8 # 7 is a way of using a mathematical expression to show its factorization, where 8 is identified as the important perfect cube factor.

• Mathematical expressions make it easier to visualize relationships between numbers.
• They are crucial for problem-solving in mathematics, providing a systematic way to approach calculations.
• Simplifying expressions through factorization is a key skill in mathematics.

By understanding mathematical expressions, you can gain deeper insights into how numbers interact and can be manipulated to solve equations or understand patterns.