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".

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, 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 the prime factors that share (including repeated primes);
3. Multiply the prime factors to get the GCF;

Example

1. $36=2\times 2\times 3\times 3$
   $54=2\times 3\times 3\times 3$
2. Notice that 2 and 3 are used in both of the factorizations?
3. However, 2 is only shared once, while 3 is shared twice.
4. So, to get the GCF, we evaluate as such:$2\times 3\times 3=18$
   Therefore, the GCF of 36 and 54 is 18.

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) multiple by both numbers.

1. Write down the prime factorisation of both numbers;
2. Write the prime factors as many times as they occur most in each of both numbers;
3. Multiply the written list of prime factors to get the LCM.

Example

1. $42=2\times 3\times 7$
   $98=2\times 7\times 7$
2. Notice that both numbers either have 2, 3, or 7 as their factor.
3. 2 occurs at most once, 3 at most once, and 7 at most twice.
4. This means we write $2\times 3\times 7\times 7$.
5. Evaluating it gives 294. Therefore, the LCM of 42 and 98 is 294.

If no prime factor occurs in both numbers (or in other words, share a prime factor), then the LCM is simply the product of both numbers.