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Numbers | Multiples of Numbers | Common Multiples | Least Common Multiple |
---|---|---|---|
2 and 3 | Multiples of 2:Multiples of 3: 2,4,6,8,10,12,… 3,6,9,12,15,…
|
6,12,18,24,… | LCM(2,3)=6 |
8 and 12 | Multiples of 8:Multiples of 12: 8,16,24,32,40,48,… 12,24,36,48,…
|
24,48,72,96,… | LCM(8,12)=24 |
A special procedure exists for finding the LCM(a,b) of a pair of numeric expressions. The LCM of two relatively prime numbers is always equal to their product.
Coprimes | LCM |
---|---|
3 and 5 | 15 |
5 and 4 | 20 |
4 and 9 | 36 |
Polynomials can also have a least common multiple. The LCM of two or more polynomials is the smallest multiple of both polynomials. In other words, the LCM is the smallest expression that can be evenly divided by each of the given polynomials.
Polynomials | Factor | LCM | Explanation |
---|---|---|---|
4x3and6xy2
|
22⋅x3and2⋅3⋅x⋅y2
|
12x3y2 | 12x3y2 is the smallest expression that is divisible by both 4x3 and 6xy2. |
3x3+3x2andx2+5x+4
|
3⋅x2⋅(x+1)and(x+1)(x+4)
|
3x2(x+1)(x+4) | 3x2(x+1)(x+4) is the smallest expression that is divisible by both 3x3+3x2 and x2+5x+4. |
Finding the LCM of polynomials requires identifying the factors with the highest power that appear in each polynomial.