What Times What Equals 48

Introduction

Have you ever stared at a multiplication problem, wondering which two numbers combine to create a specific result? The question "what times what equals 48" is more than just a simple arithmetic puzzle; it's a fundamental exploration of the building blocks of numbers. At its heart, this query asks for the factor pairs of the number 48. A factor pair consists of two whole numbers that, when multiplied together, yield the product 48. Understanding these pairs is a crucial skill that bridges basic arithmetic and more advanced mathematical concepts like algebra, number theory, and problem-solving. This article will thoroughly unpack this seemingly simple question, transforming it from a momentary curiosity into a deep appreciation for the structure and relationships within the world of numbers.

Detailed Explanation: Understanding Factors and Factor Pairs

To solve "what times what equals 48," we must first understand what a factor is. A factor of a number is a whole number that divides that number exactly, leaving no remainder. For example, 6 is a factor of 48 because 48 ÷ 6 = 8, with no leftover. When we talk about "what times what," we are specifically looking for factor pairs—two factors that are multiplied to reach the target product. Every multiplication fact you learned as a child, like 6 x 8 = 48, represents one such pair.

The process of finding all factor pairs is systematic. We begin with the smallest possible factor, which is always 1, and pair it with the number itself (48 ÷ 1 = 48). We then move to the next smallest whole number, 2. Since 48 is even, 2 is a factor (48 ÷ 2 = 24), giving us the pair (2, 24). We continue this process, checking each successive integer to see if it divides 48 evenly. It’s important to check numbers up to the square root of 48 (approximately 6.92), because after this point, we would simply be repeating pairs in reverse order (e.g., finding 8 x 6 after already having 6 x 8). This method ensures we find every unique combination without duplication.

The complete set of positive factor pairs for 48 is:

• 1 × 48 = 48
• 2 × 24 = 48
• 3 × 16 = 48
• 4 × 12 = 48
• 6 × 8 = 48

However, mathematics also acknowledges negative numbers. Since a negative times a negative equals a positive, we also have a corresponding set of negative factor pairs:

• (-1) × (-48) = 48
• (-2) × (-24) = 48
• (-3) × (-16) = 48
• (-4) × (-12) = 48
• (-6) × (-8) = 48

Thus, the full answer to "what times what equals 48" includes ten distinct ordered pairs when considering both positive and negative integers.

Step-by-Step Breakdown: A Methodical Approach

Let’s walk through the logical, step-by-step process to find all factor pairs of 48, a method applicable to any integer.

Step 1: Start with 1. The number 1 is a factor of every integer. Divide 48 by 1: 48 ÷ 1 = 48. This gives the pair (1, 48).

Step 2: Test the next integer, 2. Check divisibility. 48 is even, so it’s divisible by 2. 48 ÷ 2 = 24. Pair: (2, 24).

Step 3: Test 3. Add the digits of 48 (4+8=12). Since 12 is divisible by 3, 48 is divisible by 3. 48 ÷ 3 = 16. Pair: (3, 16).

Step 4: Test 4. A number is divisible by 4 if its last two digits form a number divisible by 4. 48 ÷ 4 = 12. Pair: (4, 12).

Step 5: Test 5. Numbers ending in 0 or 5 are divisible by 5. 48 ends in 8, so it is not divisible by 5. No pair.

Step 6: Test 6. Since 48 is divisible by both 2 and 3, it is divisible by 6. 48 ÷ 6 = 8. Pair: (6, 8).

Step 7: Determine when to stop. The next number to test is 7. 48 ÷ 7 ≈ 6.857, not a whole number. More importantly, we have now reached the square root of 48 (~6.92). Any factor larger than 6 would have already been paired with a factor smaller than 6 (e.g., the partner of 8 is 6, which we already have). Therefore, our search for unique positive pairs is complete.

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