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Factors:  When two or more integers are multiplied, each number is a factor of the product. "To factor" means to write the number or term as a product of its factors.

Examples of Factors and Proper Factors

15 = 3 x 5, so 3 and 5 are factors of 15. Also, 15 = 1 x 15, 1 and 15 are factors of 15. Therefore, 15 has four positive factors: 1, 3, 5, 15. When factors are multiplied together, the result is a product. For example, 3 x 5 results in the product 15.

Many textbooks assume that when you look for factors, you only consider the positive factors. But what about negative factors? You can multiply -1 and -15 to produce +15 and -3 and -5 to get +15. So, including both postive and negative factors, the factors of 15 are : ±1, ±3, ±5, ±15.

Many textbooks assume that proper factors are positive factors that are less than the number itself. We will also consider only positive factors when we are looking for proper factors because of our definitions of perfect numbers, deficient numbers, abundant numbers, amicable numbers. Thus, the proper factors of 15 are the positive factors of 15 that are less than 15 itself: 1, 3, and 5.

The factors of 6 are ±1, ±2, ±3, and ±6. The proper factors are 1, 2, and 3.

What are the factors of 36? (Assume we want both positve and negative factors.)

 36 = 1 x 36 36 = 2 x 18 36 = 3 x 12 36 = 4 x 9 36 = 6 x 6 36 = -1 x -36 36 = -2 x -18 36 = -3 x -12 36 = -4 x -9 36 = -6 x -6

So, the factors of 36 are ±1, ±2, ±3, ±4, ±6, ±9, ±12 ,±18, ±36. What are the proper factors of 36? Nonexample of Factors

Since 3 does not divide 4 exactly, then 3 is not a factor of 4.