Why do perfect squares have odd numbers of factors?
Answers
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Here is ur answer......✌️✌️✌️
A factor is a number that can be multiplied with another number to get a specific product. For most (not perfect square numbers), we can think of factors as coming in pairs.
For example, let's look at the factors of 12. The following multiplication facts/pairs can get a product of 12: 1X12, 2X6, and 3X4. Therefore, the factors are 1, 2, 3, 4, 6, 12. This has an even number of factors because each factor has another factor paired with it to get to the desired product.
However, the case of perfect squares are different because for one of the factors, the paired factor is itself. Let's look at 16 as an example. The following multiplication facts/pairs can get a product of 36: 1x36, 2x18, 3x12, 4x9, 6x6. Note that 6 shows up twice, but when we write the list of factors, 6 need only be mentioned once. Therefore, the factors are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Here, there is an odd number of factors because the square root of the perfect square (in this case 6) does not have a pair.
Therefore, perfect squares have an odd number of factors because the square root of the perfect square does not have a pair.