Factors

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Factors can be thought of as the multiplying building blocks for integers. A factor is an integer that divides another integer without leaving a remainder. In general, an integer x is a factor of the integer y if is also an integer.

Because 8 divides 24 evenly (as 3), with no remainder, 8 is a factor of 24. Eight factors into 2 × 2 × 2. Hence, the integer 24 is made up of the factors 2, 2, 2, and 3 multiplied together.

Every integer is a factor of itself, because it divides itself evenly, with no remainder. Also, 1 is a factor of every number. All integers therefore have factors, because each integer (except 0) has at least two factors—1 and itself.

When an integer has only two factors, then it is a prime number. Because the only factors of 5 are 1 and 5, 5 is a prime number. When a number has more than two factors, then it is a composite number. Besides 1 and 15, 15 has two more factors—3 and 5; hence, 15 is a composite number.

All even numbers, 2, 4, 6, 8,… have 2 as a factor. By definition, an even number is a multiple of 2 and can be written as 2n, where n is a positive integer. Odd numbers, 1, 3, 5, 7,… are of the form 2n + 1. Therefore, by definition, 2 cannot be a factor of an odd number.

A factor that is a prime number is called a prime factor. For instance, 3 and 5 are prime factors of 15, and 3 × 5 = 15. The "fundamental theorem of arithmetic" states that every integer can be expressed as a unique product of prime factors. In other words, every whole number can be expressed as a product of primes (and 1) unique to it. For example, the prime factors of 20 are 2, 2, and 5, since 2 × 2 × 5 = 20, and no other set of prime factors will yield the number 20.

What is the greatest common factor of 16 and 8? This question asks which factors 16 and 8 have in common and which of those factors is the greatest. For instance, 4 is a common factor of both 8 and 16. Eight is also a common factor of 8 and 16. But 8 is the greatest common factor of 8 and 16. As the name suggests, the greatest common factor of two or more numbers is the largest factor shared by them. What is the greatest common factor of 12, 8, and 4? It is 4 because 4 is the largest number that divides all three numbers.

Primes, Puzzles Of.

Amdahl, Kenn, and Jim Loats. Algebra Unplugged. Broomfield, CO: Clearwater Publishing Co., 1995.

Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: Addison-Wesley, 2001.

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