Problem 91 - Algebra A Combined Function

Question

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. $$ 16 $$

Step-by-Step Solution

Answer

16 can be written as \(8 \times 2\).

Step 1: Find Perfect Cubes Less Than 16

List the perfect cubes less than 16. These cubes include 1 (since \(1^3 = 1\)) and 8 (since \(2^3 = 8\)).

Step 2: Check Division Possibility

Check if 16 can be divided by either of these cubes to result in an integer. For 8, perform the division: \(16 \div 8 = 2\). This produces an integer.

Step 3: Verify Factorization

Ensure that 16 is equal to the product of 8 and the resulting quotient, 2. Calculate \(8 \times 2 = 16\). This confirms the factorization.

Key Concepts

Perfect Cube

Understanding what a perfect cube is can make math tasks simpler. A perfect cube is an integer that can be expressed as the cube of another integer. This means if you multiply a number by itself three times, the result is a perfect cube.

• Examples: 1, 8, 27, and 64 are all perfect cubes.
• This happens because: \(1^3 = 1\), \(2^3 = 8\), \(3^3 = 27\), and \(4^3 = 64\).

Recognizing perfect cubes in a set of numbers helps in problems where you need to express an integer as a product of two specific factors.

Division Strategy

The division strategy is a key method when searching for factor combinations. To use this strategy, you identify which smaller numbers, such as perfect cubes, divide the given integer evenly.
Start by listing all potential divisors that are perfect cubes. For example, with 16, you check 1 and 8, which are the cubes of 1 and 2, respectively. Next, perform division:

• Try dividing 16 by each cube until you find a division that results in a whole number.
• For 8, \(16 \div 8 = 2\), this result is a whole number, indicating a successful division.

This strategy is a straightforward way to decompose numbers into products of interest.

Mathematical Proof

A mathematical proof is a series of logical steps that show a statement or solution is true beyond doubt. In factorization, you prove correctness by re-multiplying factors to verify the original number.
Using the division example: Once you divide 16 by 8 and get 2, you must check this result makes sense. Multiply back the factors to see if they lead to the original number:

• \(8 \times 2 = 16\)

This confirms that our division and factor identification is correct. Proofs like these help cement our understanding and ensure accuracy.

Factor Verification

It's essential to verify factors to avoid mistakes in mathematical exercises. Verification involves checking that multiplying your identified factors returns the initial integer.
Begin with your found factors, like for 16, which splits into 8 and 2. Multiply them to test their validity:

• If \(8 \times 2 = 16\), then both numbers are indeed correct factors.
• If any discrepancy arises, recheck calculations or divisors.

Verification not only builds confidence but is also a necessary step in problem-solving to ensure correct answers.