There are several pairs of numbers that multiply to 36, encompassing both positive and negative integers. These pairs are known as the factors of 36.
Understanding Factors of 36
A factor of a number is an integer that divides the number evenly, leaving no remainder. When we talk about "two numbers that multiply to 36," we are looking for pairs of these factors whose product is exactly 36.
Positive Integer Pairs
When considering positive integers, the pairs that multiply to 36 are:
| Numbers | Pair |
|---|---|
| 1 × 36 | (1, 36) |
| 2 × 18 | (2, 18) |
| 3 × 12 | (3, 12) |
| 4 × 9 | (4, 9) |
| 6 × 6 | (6, 6) |
These are all the unique pairs of positive whole numbers that, when multiplied together, result in 36.
Negative Integer Pairs
It's also important to remember that two negative numbers multiplied together yield a positive number. Therefore, for every positive pair of factors, there is a corresponding negative pair:
- (-1, -36)
- (-2, -18)
- (-3, -12)
- (-4, -9)
- (-6, -6)
These pairs also multiply to positive 36. For example, -6 multiplied by -6 equals 36.
How to Find These Factors
To systematically find all integer factors of a number like 36:
- Start with 1: Always begin by dividing the number by 1. The result will be the number itself (1 and 36).
- Check subsequent integers: Move on to 2, 3, 4, and so on. For each integer, check if it divides 36 evenly.
- If it does, then both the divisor and the quotient form a pair of factors.
- Continue this process until the divisor you are checking is greater than the square root of the original number (for 36, the square root is 6). Once you reach or pass the square root, you will have found all unique positive pairs.
- Include negative counterparts: For every positive pair found, its negative counterpart (e.g., if (2, 18) is a pair, then (-2, -18) is also a pair) will also multiply to the positive original number.
In conclusion, the numbers that multiply to 36 include several positive integer pairs and their negative integer counterparts, offering a complete set of solutions.
