Problem 3 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The largest perfect-square _____ of 27 is 9. The largest _____ -cube factor of 16 is 8.

Step-by-Step Solution

Answer

Perfect square: 'factor'; perfect cube: 'perfect'. 9 and 8.

Step 1: Identify perfect squares

First, list the perfect squares that are less than or equal to 27. The perfect squares are numbers like 1, 4, 9, 16, and 25.

Step 2: Find the largest perfect square factor

Check each listed perfect square from the previous step to see if it divides 27 evenly (i.e., the result of the division is an integer). The largest perfect square that evenly divides 27 is 9.

Step 3: Identify perfect cubes

Now, list the perfect cubes that are less than or equal to 16. The perfect cubes are numbers like 1 and 8.

Step 4: Find the largest perfect cube factor

Check each listed perfect cube from the previous step to see if it divides 16 evenly. The largest perfect cube that divides 16 evenly is 8.

Key Concepts

Factor

In mathematics, a "factor" is an integer that can be multiplied by another integer to produce a specific number. Factors are integral to understanding the divisibility and structure of numbers.
Let's take an example: if we have the number 27, its factors include 1, 3, 9, and 27, because all these numbers can be multiplied together in various combinations to equal 27.
This concept helps determine relationships between numbers and is essential in simplifying mathematical expressions. Handling factors particularly helps in solving equations, simplifying fractions, and working with algebraic expressions.

Perfect Cube

A "perfect cube" is a number that can be expressed as the cube of an integer. This means if you multiply an integer by itself twice (including 1 as itself), you'll end up with a perfect cube.
For instance, 8 is a perfect cube because it equals 2 multiplied by itself twice:

• 2 \( \times \) 2 \( \times \) 2 = 8

Identifying perfect cubes is useful in many areas of mathematics.
For example, in the original problem, the largest perfect cube factor of 16 is 8. Understanding perfect cubes can greatly aid in factoring larger expressions, recognizing patterns in numbers, and solving polynomial equations, where cube roots might be necessary.

Integer Division

Integer division is the division in which the resulting quotient is an integer, with the remainder being discarded and typically being 0 for clean division without a remainder.
This form of division is fundamental in determining the divisibility of numbers. To ascertain if one number divides another without any leftover, integer division is applied.
For example, when we examine whether 9 is a factor of 27, we divide 27 by 9 using integer division. The result is exactly 3 with no remainder, confirming 9 as a factor. Similarly, 16 divided by 8 yields 2.
In mathematics, understanding integer division helps identify factors quickly and is a crucial skill when working on problems involving divisibility, simplifying equations, and performing algorithmic operations where precise quotients are needed without fractions.