Arithmetic/More About Multiplication

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Factors and Multiples[edit | edit source]

Is One a Factor of Everything?[edit | edit source]

To answer the question above, yes. 1 is a factor of any number n. For proof, consider this statement: "any number n is a factor of itself when multiplied by one".

let n = any number, then any number times 1 is itself. This picture also demonstrates the commutative property of multiplication which basically means the operation of multiplication can be performed in any order.

Proof: Let n = 27 then 27 * 1 = 27 because any number multiplied by 1 is itself. The reason for this is say you had a bag of 12 marbles. If you had 2 bags of marbles you would have 24 marbles because 12 is multiplied by the number 2 because you have 2 bags. However, if you only have 1 bag of 12 marbles, you only have 12 marbles. Therefore, we know this statement to be a fact: Any number n can be factored to n * 1.

58 million times 1? Answer: 58 million. It does not matter how big the number(or even how small, as in this example: 1*(-245) = -245.) Though, negative numbers are beyond the scope of this page, as you continue to increase your understanding of mathematics, you will learn of negative numbers and absolute value.

Is Zero a Factor of Anything?[edit | edit source]

To answer the question above, no.

A number is a factor of another number if it can be multiplied by a whole number to give the number is it a factor of. Anything multiplied by zero is also zero, which means that the only number that could possibly have zero as a factor is zero itself.

Phrased differently, the multiples of 0 go 0, 0, 0, 0... and so on, never becoming any larger than 0. Since a factor is the reverse of a multiple, there are no numbers other than 0 with a factor of 0 as well.

A Look Ahead at Prime Numbers[edit | edit source]

The Greatest Common Factor[edit | edit source]

The Greatest common Factor (GCF) of two whole numbers is the highest number that can be divided by both of the original numbers to get a whole number result.

Finding the GCF of a number[edit | edit source]

  1. Write down to prime factorization of the two numbers.
  2. Find prime factors that are the same.
  3. Multiply the same prime factors together for the GCF.

Example[edit | edit source]

  1. Notice that two and seven are used in both the equations?

  2. Therefore, the GCF of 28 and 98 is 14.

The Least Common Multiple[edit | edit source]