When faced with the question, "What times what equals 125?" many people pause, unsure of how to approach the problem. Whether you're solving a math problem for school, calculating dimensions for a project, or simply satisfying your curiosity, understanding how to break down this type of question is essential. This guide will walk you through everything you need to know, from the basics of multiplication to advanced methods for finding factors. By the end, you'll not only know the answer but also understand how to approach similar problems confidently in the future.
At its core, the question "What times what equals 125?" is about finding two numbers that, when multiplied together, result in 125. This is a problem of factoring, and factors are the building blocks of numbers. For 125, the solution is straightforward once you understand the math behind it. But what if you’re dealing with a larger or more complex number? Or what if you're trying to solve this without a calculator? This guide will provide step-by-step solutions, common pitfalls to avoid, and practical tips for tackling such problems both now and in the future.
Quick Reference
- Immediate action: Start by dividing 125 by smaller numbers like 1, 5, or 25 to find factors.
- Essential tip: Remember that 125 = 5 × 5 × 5, or 5³, which means it's a cube of 5.
- Common mistake: Don't forget to check for both positive and negative factors (e.g., -5 × -25 = 125).
Step-by-Step Guide to Solving "What Times What Equals 125"
Step 1: Understand the Problem
To solve “What times what equals 125?” you need to find pairs of numbers, called factors, that multiply together to give 125. For example, in the equation a × b = 125, both a and b are factors of 125. The goal is to figure out all possible values for a and b.
Remember, multiplication involves two numbers. If one number is known, the other can be calculated by dividing 125 by the known factor. For example, if you know that 5 is a factor, the other number can be found by calculating 125 ÷ 5 = 25.
Step 2: Start with the Smallest Factors
Every number has at least two factors: 1 and itself. For 125, this means:
- 1 × 125 = 125
Next, check smaller numbers to see if they divide evenly into 125. For example:
- 125 ÷ 2 = 62.5 (not a whole number, so 2 is not a factor)
- 125 ÷ 3 = 41.67 (not a whole number, so 3 is not a factor)
- 125 ÷ 5 = 25 (a whole number, so 5 is a factor)
Now we know another pair: 5 × 25 = 125. So far, the factors are:
- 1 × 125
- 5 × 25
Step 3: Consider Negative Factors
Multiplication allows for both positive and negative numbers. For example:
- -1 × -125 = 125
- -5 × -25 = 125
This means the complete list of factor pairs for 125 includes both positive and negative values:
- 1 × 125
- 5 × 25
- -1 × -125
- -5 × -25
Step 4: Recognize Prime Factors
Prime factorization is breaking down a number into its simplest "building blocks." For 125, the prime factor is 5. Here's how you find it:
- Start with the smallest prime number, 2. Since 125 is odd, it’s not divisible by 2.
- Next, try 3. 125 ÷ 3 = 41.67 (not a whole number).
- Try 5. 125 ÷ 5 = 25 (a whole number).
- Now repeat the process for 25. 25 ÷ 5 = 5.
This shows that 125 can be expressed as 5 × 5 × 5, or 5³. Understanding prime factorization helps when solving more complex equations.
Real-World Applications of "What Times What Equals 125"
Application 1: Solving Math Problems
Students often encounter questions like this in algebra, geometry, or number theory. For instance, you might need to find factors when simplifying fractions, solving equations, or working with area calculations. Knowing how to break down a number like 125 into its factors is a foundational skill.
Application 2: Everyday Problem Solving
Outside of school, understanding factors can help in practical scenarios. For example, if you’re building a rectangular garden with an area of 125 square feet, you’ll need to find dimensions that multiply to 125. Possible dimensions include:
- 1 ft × 125 ft
- 5 ft × 25 ft
Knowing the factors of 125 allows you to choose a layout that fits your available space.
Application 3: Financial Calculations
Factors also come into play in financial scenarios. For example, if you’re dividing 125 among a group of people, you might want to know all the possible ways to split it evenly. By identifying the factors of 125, you can determine group sizes that work (e.g., 5 people each get 25).
Tips and Tricks for Solving Similar Problems
Tip 1: Use Division to Test Factors
Always start by dividing the number by smaller integers to see if they result in whole numbers. If they do, you’ve found a factor pair.
Tip 2: Memorize Key Multiplication Facts
Knowing multiplication tables up to 12 × 12 can save time when solving problems like this. For example, recognizing that 5 × 25 = 125 comes naturally if you’re familiar with these facts.
Tip 3: Leverage Prime Factorization
Breaking a number down into its prime factors simplifies the process of finding all factor pairs. For 125, recognizing that it’s 5³ immediately points to 5 as a key factor.
Tip 4: Double-Check Your Work
After identifying factor pairs, multiply them to ensure they equal the original number. This step helps you catch errors and confirm accuracy.
What are all the factor pairs of 125?
The factor pairs of 125 are: (1, 125), (5, 25), (-1, -125), and (-5, -25).
Is 125 a prime number?
No, 125 is not a prime number because it has more than two factors. Its prime factorization is 5 × 5 × 5, or 5³.
How can I solve similar problems for larger numbers?
For larger numbers, use prime factorization to break them into smaller components. Test divisibility starting with smaller primes like 2, 3, 5, and so on. Use a calculator for efficiency if needed.