This guide explores the various ways to find the answer to "what times what equals...?" Whether you're tackling simple multiplication problems or more complex equations, we'll cover the methods and tools available to help you solve them. We'll even delve into the different scenarios you might encounter and how to approach each one effectively.
What is a "What Times What Equals" Problem?
Essentially, this phrasing describes a multiplication problem where you're given the product (the result of multiplication) and need to find the factors (the numbers that were multiplied together). For example, "What times what equals 12?" is asking you to find two numbers that multiply to give you 12.
How to Solve "What Times What Equals..." Problems
There are several ways to approach these types of problems:
1. Mental Math and Factoring
For smaller numbers, mental math is often the quickest method. You can simply list the pairs of numbers that multiply to give the desired product. For example, for "What times what equals 12?", you might think:
- 1 x 12 = 12
- 2 x 6 = 12
- 3 x 4 = 12
Therefore, the answer is 1 and 12, 2 and 6, or 3 and 4.
2. Prime Factorization
For larger numbers, prime factorization can be helpful. This involves breaking down the number into its prime factors (numbers divisible only by 1 and themselves). Let's say the question is "What times what equals 72?".
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Find the prime factors of 72: 72 = 2 x 2 x 2 x 3 x 3 = 2³ x 3²
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Now, you can combine these prime factors in different ways to find pairs of factors:
- 2 x 36 = 72
- 3 x 24 = 72
- 4 x 18 = 72
- 6 x 12 = 72
- 8 x 9 = 72
3. Using a Calculator
For very large numbers or when you need a quick solution, a standard calculator is a great tool. However, it won't directly give you the factor pairs. You'll need to use trial and error or other methods to find the factors. Many scientific calculators have features that can help with factoring, but that's often beyond the scope of a simple "what times what equals" question.
4. Online Calculators
Numerous online calculators are specifically designed to find factors. These tools often provide a comprehensive list of factor pairs, making them very useful for larger numbers. Simply input the desired product, and the calculator will give you all the possible combinations.
What if it's a More Complex Equation?
The phrase "what times what" can sometimes be used in a broader context, referring to more complex algebraic equations. For example, "What times what equals x² + 6x + 9?" This would involve factoring a quadratic expression, requiring a deeper understanding of algebra.
What if I Need to Find Three or More Factors?
The problem can also involve finding more than two numbers whose product is a given number. This requires a more systematic approach and understanding of number theory.
Frequently Asked Questions
How do I find factors quickly?
For smaller numbers, mental math and memorizing multiplication tables are the fastest methods. For larger numbers, online calculators or prime factorization can help.
What are the factors of a number?
Factors are numbers that divide evenly into a larger number without leaving a remainder.
Are there any online tools to help with finding factors?
Yes, many websites offer online calculators that find the factors of a given number. Search for "factor calculator" to find one.
Can I use a calculator to find factors?
A standard calculator won't directly provide the factors. However, you can use it to check if a particular number is a factor by performing division.
By using the methods described above, you can efficiently solve any "what times what equals" problem, whether it involves simple multiplication or more complex algebraic concepts. Remember to choose the method best suited to the size and complexity of the number you are working with.
