What Times What Equals 90

What Times What Equals 90? Exploring the Factors and Applications of 90

Finding the factors of a number, like solving "what times what equals 90," might seem like a simple arithmetic problem. However, understanding this seemingly basic concept unlocks a broader understanding of mathematics, including prime factorization, divisibility rules, and even the foundations of algebra. This article will delve into the multiple solutions to this equation, explore the mathematical concepts involved, and demonstrate its relevance in various applications.

Introduction: Unpacking the Problem

The question "what times what equals 90?" asks us to identify pairs of numbers whose product is 90. This involves finding the factors of 90. Factors are numbers that divide evenly into a given number without leaving a remainder. This seemingly simple question opens doors to understanding fundamental mathematical concepts crucial for more advanced studies.

Finding the Factors of 90: A Step-by-Step Approach

Let's systematically find all the pairs of numbers that multiply to 90. We can start by listing the factors in ascending order:

1. 1 x 90: This is the most straightforward pair. Any number is divisible by 1 and itself.

2. 2 x 45: Since 90 is an even number, it's divisible by 2.

3. 3 x 30: 90 is divisible by 3 because the sum of its digits (9+0=9) is divisible by 3.

4. 5 x 18: 90 ends in 0, making it divisible by 5.

5. 6 x 15: 90 is divisible by both 2 and 3, therefore it's divisible by 6 (2 x 3).

6. 9 x 10: 90 is divisible by 9 because the sum of its digits (9) is divisible by 9.

These are all the pairs of integer factors of 90. Note that we've now exhausted all possibilities. If we were to continue checking larger numbers, we would simply find the reverse of the pairs already listed (e.g., 10 x 9, 15 x 6, etc.).

Prime Factorization: The Building Blocks of Numbers

The factors of 90 can also be analyzed using prime factorization. Prime factorization is the process of expressing a number as a product of its prime factors. Prime numbers are numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.).

To find the prime factorization of 90, we can use a factor tree:

90 = 2 x 45 45 = 3 x 15 15 = 3 x 5

Therefore, the prime factorization of 90 is 2 x 3 x 3 x 5, or 2 x 3² x 5. This representation shows the fundamental building blocks of 90. Every other factor of 90 is a combination of these prime factors. Understanding prime factorization is fundamental in various areas of mathematics, such as simplifying fractions and solving algebraic equations.

Beyond Integers: Exploring Rational and Real Numbers

Our exploration so far has focused on integer factors. However, the equation "what times what equals 90" can also have solutions involving rational numbers (fractions) and even real numbers (including irrational numbers).

For example:

• (1/2) x 180 = 90: This shows that we can use fractions to achieve the desired product.

• (1/3) x 270 = 90: More examples with fractions are easily generated.

• √90 x √90 = 90: Here, we're using the square root of 90, which is an irrational number (it cannot be expressed as a simple fraction). Irrational numbers are real numbers, and they can also satisfy the equation.

The possibilities expand considerably when we consider non-integer numbers.

Applications of Finding Factors: Real-World Examples

Understanding factors and their relationships goes beyond theoretical mathematics. Here are some practical applications:

• Area Calculation: If the area of a rectangle is 90 square units, finding the factors helps determine the possible dimensions of the rectangle (length and width).

• Array Arrangement: If you have 90 objects and want to arrange them in a rectangular array, the factors determine the possible number of rows and columns.

• Division Problems: Finding factors is crucial for simplifying division problems and understanding divisibility rules.

• Algebraic Equations: Factoring numbers is a fundamental skill needed to solve quadratic and other polynomial equations in algebra.

• Coding and Programming: In computer science, understanding factors and prime factorization is important for various algorithms and data structures.

Divisibility Rules and Their Role in Factor Finding

Learning divisibility rules can significantly speed up the process of finding factors. Here are some relevant divisibility rules for 90:

• Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8). 90 is divisible by 2.

• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. (9+0=9, which is divisible by 3). Therefore, 90 is divisible by 3.

• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5. 90 is divisible by 5.

• Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3. Since 90 is divisible by both, it is divisible by 6.

• Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9. (9+0=9, which is divisible by 9). Therefore, 90 is divisible by 9.

• Divisibility by 10: A number is divisible by 10 if its last digit is 0. 90 is divisible by 10.

These rules provide a quick way to check for potential factors, making the process more efficient.

Frequently Asked Questions (FAQ)

Q: Are there any negative factors of 90?

A: Yes. Since a negative number multiplied by a negative number results in a positive number, we can also have pairs like (-1) x (-90), (-2) x (-45), and so on.

Q: Can decimals be used as factors of 90?

A: Yes, as mentioned earlier, we can use decimals or fractions to find numbers that multiply to 90. The possibilities become infinite when we extend beyond integers.

Q: How can I use this knowledge to solve more complex problems?

A: This understanding of factors forms the basis for solving more complex mathematical problems involving equations, algebraic manipulations, and geometric calculations. It's a fundamental concept that builds upon itself.

Q: Is there only one correct answer to “What times what equals 90?”

A: No. There are multiple correct answers, depending on whether you are limiting your answers to integers, or expanding it to include fractions and decimals. There are also multiple pairs of integers, and their negative counterparts.

Conclusion: The Enduring Significance of Simple Arithmetic

The seemingly simple question, "what times what equals 90," provides a gateway to understanding fundamental mathematical concepts like factors, prime factorization, divisibility rules, and the broader application of these concepts in various fields. Mastering the ability to find the factors of a number is a crucial building block for more advanced mathematical studies and problem-solving in diverse real-world scenarios. From calculating areas to solving equations, the skill of factoring plays a significant role in quantitative reasoning and mathematical fluency. While this article focused on 90, the methods and principles discussed can be applied to any number, showcasing the universality and importance of these core mathematical concepts.