Factor pairs are two numbers that multiply together to give a specific product. For example, 2 and 3 are a factor pair of 6 because 2 times 3 equals 6. We can visualize this with rectangles where the dimensions represent the factors and the area represents the product.
To find all factors systematically, we test divisibility starting from 1. We check if numbers divide evenly into our target number. For 48, we start with 1: 48 divided by 1 equals 48. Then 2: 48 divided by 2 equals 24. We continue this process. Importantly, we only need to check up to the square root of the target number, which for 48 is approximately 6.93, so we only need to check up to 6.
Now let's find all factors of 48 by testing divisibility. We start with 1: 48 divided by 1 equals 48, so both 1 and 48 are factors. Next, 2: 48 divided by 2 equals 24, so 2 and 24 are factors. Then 3: 48 divided by 3 equals 16, giving us factors 3 and 16. Continuing with 4: 48 divided by 4 equals 12, so 4 and 12 are factors. For 5: 48 divided by 5 equals 9.6, which is not a whole number, so 5 is not a factor. For 6: 48 divided by 6 equals 8, so 6 and 8 are factors. Numbers 7 and 8 give non-whole or repeated results. Our complete list of factors is: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
Now we pair the factors to form factor pairs. We take our list of factors and pair them from opposite ends. The first factor 1 pairs with the last factor 48. The second factor 2 pairs with the second-to-last factor 24. Continuing this pattern: 3 pairs with 16, 4 pairs with 12, and 6 pairs with 8. We can verify each pair by multiplying: 1 times 48 equals 48, 2 times 24 equals 48, 3 times 16 equals 48, 4 times 12 equals 48, and 6 times 8 equals 48. This gives us our complete set of factor pairs.
We can visualize each factor pair as a rectangle with area 48. The first pair (1,48) creates a very long, thin rectangle. The pair (2,24) creates a less extreme rectangle. The pair (3,16) is more balanced. The pair (4,12) is even more square-like. And the pair (6,8) creates the most square-like rectangle. All these rectangles have the same area of 48 square units. We can also verify our factor pairs using prime factorization. The prime factorization of 48 is 2 to the power of 4 times 3 to the power of 1. To find the total number of factors, we add 1 to each exponent and multiply: (4+1) times (1+1) equals 10 factors. Since factors come in pairs, we have 10 divided by 2 equals 5 factor pairs, which matches our result.
Here is our complete solution summary. The factor pairs of 48 are: (1, 48), (2, 24), (3, 16), (4, 12), and (6, 8). To find factor pairs of any number, follow this method: First, test divisibility starting from 1. Second, only check up to the square root of the number. Third, for each divisor you find, include its corresponding pair. Fourth, verify your results by multiplication. Factor pairs have practical applications in many areas. They help when arranging objects in rectangular patterns, finding dimensions of areas, and solving algebraic equations. Understanding factor pairs is a fundamental mathematical skill with wide-ranging uses.