Multiples of numbers in Math, are the results from multiplying a given number by integers.
Integers = { …. , -3 , -2 , -1 , 0 , 1 , 2 , 3 , …. }
Numbers and Multiples
We can look at an example of the number 8.8 × 0 = 0 , 8 × 1 = 8 , 8 × 2 = 16 , 8 × 3 = 24
The numbers 0, 8, 16, and 24 are all multiples of the number 8.
They are all the result of multiplying 8 by another whole number.
Fractions and Multiples
It’s the case that fractions can also have multiples like whole numbers do.
{\large{\frac{1}{3}}} \times 0 \space = \space 0 , {\large{\frac{1}{3}}} \times 1 \space = \space {\large{\frac{1}{3}}} , {\large{\frac{1}{3}}} \times 2 \space = \space {\large{\frac{2}{3}}} , {\large{\frac{1}{3}}} \times 3 \space = \space 1
All the answers above, are multiples of \frac{1}{3}.
Multiples and Numbers, Common Multiples
With numbers and multiples, a common multiple is a number that is a multiple of 2 or more other numbers.
We can list some multiples of 4. => 4 , 8 , 12 , 16 , 20 , 24
Some multiples of 6. => 6 , 12 , 18 , 24
It can be seen that 12 and 24 are common multiples of 4 and 6.
Lowest Common Multiple
What are lowest common multiples?
The lowest common multiple of a group of 3 positive numbers or a larger, is the smallest positive number that is a multiple of each number.
Examples
(1.1)
Find the lowest common multiple of the numbers 5 and 8.
Solution
One effective method is to list out the multiples of each number, and looking for the smallest shared multiple in both lists.
5 => { 5 , 10 , 15 , 20 , 25 , 30 , 35 , 40 , 45 , 50 , …. }
8 => { 8 , 16 , 24 , 32 , 40 , 48 , 56 , 64 , 72 , 80 , …. }
From looking at the lists it can be seen that 40 is the lowest common multiple of 5 and 8.
(1.2)
Find the lowest common multiple of 3, 6 and 12.
Solution
3 => { 3 , 6 , 9 , 12 , 15 , 18 , 21 , 24 , 27 , 30 , 33 , 36 , 39 , 42 , 45 , 48 , … }
6 => { 6 , 12 , 18 , 24 , 30 , 36 , 42 , 48 , 54 , … }
12 => { 12 , 24 , 36 , 48 , … }
48 is the lowest common multiple of 3, 6 and 12.
(1.3)
There is another approach than can also be used to find a lowest common multiple.
We begin by drawing up an appropriate table, and we then divide each given number exactly by prime numbers. starting with the smallest we can.
We keep dividing exactly by prime numbers where we can, increasing the value of the dividing prime number if necessary.
Until we eventually obtain a result of 1 for each entry,and there is no more exact division to be carried out.
We can look at the case of the lowest common multiple of 4 and 22.
The prime number we are dividing by is written on the left of 4 and 22, and the results of the division are written below each number.
Any number that can’t be divided exactly by the prime number is left as it sits.
The prime numbers that were used for each exact division, multiplied together, will give the lowest common multiple of the numbers 4 and 22.
2 × 2 × 11 = 44
This can also be seen by drawing up a list of the factors of 4 and 22.
4 => { 4 , 8 , 12 , 16 , ……. , 36 , 40 , 44 , 48 , … }
22 => { 22 , 44 , 66 , 88 , … }
(1.4)
Find the lowest common multiple of 20 and 161.
Solution
Here we can use the table and prime number division method.
2 was used in the division twice, 5, 7 and 23 one time.
2 × 2 × 5 × 7 × 23 × 57 = 3’220
3’220 is the lowest common multiple of 20 and 161.
(1.5)
Find the lowest common multiple of 4, 8 and 18.
Solution
We can also aim to find the lowest common multiple of more than just two numbers.
The table and prime number division method can still work the same way.
2 was used in the division three times, and 9 was used one time.
2 × 2 × 2 × 9 = 44
44 is the lowest common multiple of 4, 8 and 18.