lowest common multiple (LCM) – of a set of numbers is the smallest positive number that every number divides equally into exactly
Small example
Find Lowest Common Multiple (LMC) of 4 and 6.
4 multiplied is 4 8 12 16 20
6 multiplied is 6 12 18 24
LMC of 4 and 6 is 12 i.e. 12 is the smallest multiple they have in common.
For more difficult problems carry out the following process.
1. First carry out prime factorisation of each number (prime factorisation).
2. Put them in matching columns.
3. Multiply every column together.
Note: The memory problem is to remember what to multiply when you are asked for the highest common factor or the lowest common multiplier.
To remember that with a LMC question you multiply every prime factor in each column, use the following mnemonic:
Little cows moo (LCM) on top of each column.
HCF = Multiply same number
LMC = Multiply every column
Example 1
Find the highest common factor of prime factor 12;
`12 divide 2 = 6`
`6 divide 2 = 3`
`3 divide 3 = 1`
Summary `2 times 2 times 3`
Prime factor of 18;
`18 divide 2 = 9`
`9 divide 3 = 3`
`3 divide 3 = 1`
Summary `2 times 3 times 3`
Put them in matching columns.
Multiply every column together.
`2 times 2 times 3 times 3 = 2 times 9 = 36`
The LCM of 12 and 18 is 36.
Example 2
Find the LMC of the following three sets of numbers, 144, 180 and 352.
Prime factor of 144;
`144 divide 2 = 72`
`72 divide 2 = 36`
`36 divide 2 = 18`
`18 divide 2 = 9`
`9 divide 3 = 3`
`3 divide 3 = 1`
Summary `2 times 2 times 2 times 2 times 3 times 3 = 144`
Prime factor of 180;
`180 divide 2 = 90`
`90 divide 2 = 45`
`45 divide 3 = 15`
`15 divide 3 = 5`
`5 divide 5 = 1`
Summary `2 times 2 times 3 times 3 times 5 = 180`
Prime factor of 252;
`252 divide 2 = 126`
`126 divide 2 = 63`
`63 divide 3 = 21`
`21 divide 3 = 7`
`7 divide 7 = 1`
Summary `2 times 2 times 3 times 3 times 7 = 252`
Put them in matching columns.
Multiply every column together:
`2 times 2 times 2 times 2 times 3 times 3 times 5 times 7 = 5040`
The LCF of 144, 180 and 252 = 5040.
Example 3
Find the LCM of the following three sets of numbers, 96, 900 and 630.
Prime factor of 96;
`96 divide 2 = 48`
`48 divide 2 = 24`
`24 divide 2 = 12`
`12 divide 2 = 6`
`6 divide 2 = 3`
`3 divide 3 = 1`
Summary `2 times 2 times 2 times 2 times 2 times 3 = 96`
Prime factor of 900;
`900 divide 2 = 450`
`450 divide 2 = 225`
`225 divide 3 = 75`
`75 divide 3 = 25`
`25 divide 5 = 5`
`5 divide 5 = 1`
Summary `2 times 2 times 3 times 3 times 5 times 5 = 96`
Prime factor of 630;
`630 divide 2 = 315`
`315 divide 3 = 105`
`105 divide 3 = 35`
`35 divide 5 = 7`
`7 divide 7 = 1`
Summary `2 times 3 times 3 times 5 times 7 = 630`
Put them in columns.
Multiply every column together
= `2 times 2 times 2 times 2 times 2 times 3 times 3 times 5 times 5 times 7`
= `32 times 9 times 25 times 7`
= `50,400`
The LCM of 96, 900 and 630 = 50,400
