Arithmetic/More About Multiplication

Is One a Factor of Everything?
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".

Proof. Let n = 27, without loss of generality (sometimes written "WLOG", for short. Here, it means that you can choose whatever integer you want). Then, 27 * 1 = 27 since any number multiplied by 1 is itself.

Let's say you had a bag of 12 marbles. If you now have 2 bags of marbles, you have 24 marbles because each bag has 12 marbles (this is 2 * 12 = 24). However, if you only have 1 bag, you only have 12 marbles due to the same reasoning as before, that is, 1 * 12 = 12. Therefore, we conclude that any number n can be factored to n * 1.

58 million times 1? Answer: 58 million. The size of the number doesn't matter (as in this example: 1*(-245) = -245.) Though, negative numbers are beyond the scope of this page, but as you continue to increase your understanding of mathematics, you will learn of negative numbers and absolute value.

Is Zero a Factor of Anything?
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.

The Greatest Common Factor
The Greatest Common Factor (GCF) of two whole numbers, is the highest whole number that divides both original numbers.

Finding the GCF of a number

 * 1) Write down the prime factorisation of both numbers;
 * 2) Find all equal prime factors;
 * 3) Multiply the prime factors to get the GCF;

Example
$$98 = 2 \times 7 \times 7$$ Therefore, the GCF of 28 and 98 is 14.
 * 1) $$28 = 2 \times 2 \times 7$$
 * 1) Notice that two and seven are used in both the equations?
 * 2) $$2 \times 7 = 14$$

If both numbers don't have equal prime factors, then the GCF is 1 (since every whole number is a multiple of 1, that is, n = n * 1, as we saw previously).

The Least Common Multiple
As the name implies, the Least Common Multiple (LCM) of two whole numbers, is the smallest (least) shared (common) factor (multiple) by both numbers.


 * 1) Write down the prime factorisation of both numbers;
 * 2) Find the smallest prime factor for each one of them;
 * 3) If they are equal, then the LCM is equal to that factor;

Example
$$98 = 2 \times 7 \times 7$$
 * 1) $$28 = 2 \times 2 \times 7$$
 * 1) Notice that both numbers have 2 as their smallest prime factor.
 * 2) Therefore, the LCM of 28 and 98 is 2.

If all prime factors are different, then the LCM is 1 (again, because every whole number is a multiple of 1).