LCM and GCF
Purplemath
To find either the Least Common Multiple (LCM) or Greatest Common Factor (GCF) of two numbers, you always start out the same way: you find the prime factorizations of the two numbers.
Then (here's the trick!) you put the factors into a nice neat grid of rows and columns, compare and contrast, and then, from the table, take only what you need.
For instance:
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Find the GCF and LCM of 2940 and 3150.
First, I need to factor each of the numbers they've given me:
I started by dividing 2940 by the smallest prime that would fit into it, being 2. This left me with another even number, 1470, so I divided by 2 again. The result, 735, is divisible by 5, but 3 divides in also, and it's smaller, so I divided by 3 to get 245. This is not divisible by 3 but is divisible by 5, so I divided by 5 and got 49, which is divisible by 7.
Now I'll apply the same sequential-division process to 3150:
I've divided each of the given numbers by the smallest primes that fit into them, until I ended up with a prime result. The factorizations can be read from the numbers along the outside of the sequential divisions. So my prime factorizations are:
2940 = 2 × 2 × 3 × 5 × 7 × 7
3150 = 2 × 3 × 3 × 5 × 5 × 7
I will write these factors out, all nice and neat, with the factors lined up according to occurrance:
2940: 2×2×3 ×5 ×7×7 3150: 2 ×3×3×5×5×7
This orderly listing, with each factor having its own column, will do most of the work for me.
The Greatest Common Factor, the GCF, is the biggest ("greatest") number that will divide into (that is, the largest number that is a factor of) both 2940 and 3150. In other words, it's the number that contains all the factors common to both numbers. In this case, the GCF is the product of all the factors that 2940 and 3150 have in common.
Looking at the nice neat listing, I can see that the numbers both have a factor of 2; 2940 has a second copy of the factor 2, but 3150 does not, so I can only count the one copy toward my GCF. The numbers also share one copy of 3, one copy of 5, and one copy of 7.
2940: 2×2×3 ×5 ×7×7 3150: 2 ×3×3×5×5×7 ----:---------------- GCF: 2 ×3 ×5 ×7 = 210
Then the GCF is 2 × 3 × 5 × 7 = 210.
On the other hand, the Least Common Multiple, the LCM, is the smallest ("least") number that both 2940 and 3150 will divide into. That is, it is the smallest number that contains both 2940 and 3150 as factors, the smallest number that is a multiple of both these values; it is the multiple common to the two values. Therefore, it will be the smallest number that contains every factor in these two numbers.
Looking back at the listing, I see that 3150 has one copy of the factor of 2; 2940 has two copies. Since the LCM must contain all factors of each number, the LCM must contain both copies of 2. However, to avoid overduplication, the LCM does not need three copies, because neither 2940 nor 3150 contains three copies.
This over-duplication issue with factors often causes confusion, so let's spend a little extra time on this. Consider two smaller numbers, 4 and 8, and their LCM. The number 4 factors as 2 × 2; 8 factors as 2 × 2 × 2. The LCM needs only have three copies of 2, in order to be divisible by both 4 and 8. That is, the LCM is 8. You do not need to take the three copies of 2 from the 8, and then throw in two extra copies from the 4. This would give you 32. While 32 is a common multiple, because 4 and 8 both divide evenly into 32, 32 is not the LEAST (smallest) common multiple, because you'd have over-duplicated the 2s when you threw in the extra copies from the 4.
Let me stress again: let the nice neat listing keep track of things for you, especially when the numbers get big. Returning to the exercise:
So, my LCM of 2940 and 3150 must contain both copies of the factor 2. By the same reasoning, the LCM must contain both copies of 3, both copies of 5, and both copies of 7:
2940: 2×2×3 ×5 ×7×7 3150: 2 ×3×3×5×5×7 ----:---------------- LCM: 2×2×3×3×5×5×7×7 = 44,100
Then the LCM is 2 × 2 × 3 × 3 × 5 × 5 × 7 × 7 = 44,100.
By using this "factor" method of listing the prime factors neatly in a table, you can always easily find the LCM and GCF. Completely factor the numbers you are given, list the factors neatly with only one factor for each column (you can have 2s columns, 3s columns, etc, but a 3 would never go in a 2s column), and then carry the needed factors down to the bottom row.
For the GCF, you carry down only those factors that all the listings share; for the LCM, you carry down all the factors, regardless of how many or few values contained that factor in their listings.
Note: There is another method for finding LCMs and GCFs; it's called the "listing" method. For instance, to find the LCM of 4 and 6, you'd list their multiples, starting with the smallest and working your way up, until you found the first duplicate. This first duplicate multiple would be your LCM:
4: 4, 8, 12, 16, 20, 24, 28, 32, ... 6: 6, 12, 18, 24, 30, 36, ...
The first duplicate is 12, so this is the LCM.
But this "listing" method would be awful for large numbers like what we just did above. Would you want to try listing multiples for 2940 and 3150? I know I wouldn't. The process for GCFs can be even more painful! The factor method we used above is much easier.
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Find the LCM and GCF of 27, 90, and 84.
First, I need to find the prime factorizations:
Then I will list these factorizations neatly:
27: 3×3×3 90: 2 ×3×3 ×5 84: 2×2×3 ×7
Then the GCF (being the product of the shared factors) and the LCM (being the product of all factors) are given by:
GCF: 3 = 3 ---:-------------- 27: 3×3×3 90: 2 ×3×3 ×5 84: 2×2×3 ×7 ---:-------------- LCM: 2×2×3×3×3×5×7 = 3,780
Then the GCF is 3 and the LCM is 3,780.
By the way, if you prefer (or if you're lazy, like me), you can omit the "times" signs in your tables, and just list the factors. It'll look like this:
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Find the GCF and LCM of 3, 6, and 8.
First I factor the numbers and list their prime factorizations:
3: 3 6: 2 3 8: 2 2 2
Then my GCF and LCM are given by:
GCF: = 1 ---:-------- 3: 3 6: 2 3 8: 2 2 2 ---:-------- LCM: 2 2 2 3 = 24
Note that 3, 6, and 8 share no common factors. While 3 and 6 share a factor, and 6 and 8 share a factor, there is no prime factor that all three of them share. Since 1 divides into everything, then the greatest common factor in this case is just 1. When 1 is the GCF, the numbers are said to be "relatively" prime; that is, they are prime, relative to each other.
Then the GCF is 1 and the LCM is 2 × 2 × 2 × 3 = 24.
The GCF doesn't come up that much at this stage in mathematics, though some books use it for factoring polynomial expressions by having the student find the GCF of all the terms in the polynomial and divide this value out of every term. But the LCM comes up every time you need to find a common denominator for fractions. The factor technique I demonstrated above works even for polynomial fractions. (The other method for finding the LCM, the "listing" method, will not work for polynomials, which is why you need to learn the factor method.) If you need to find the LCM of two (or more) polynomials, you can do the exact same procedure as above:
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Find the LCM of x3 + 5x2 + 6x and 2x3 + 4x2.
First I factor the polynomials: x3 + 5x2 + 6x = x(x2 + 5x + 6) = x(x + 2)(x + 3), and 2x3 + 4x2 = 2x2(x + 2). Then I list these factors out, nice and neat:
Then the LCM is the product of one entry from each column:
I take two copies of "x", because 2x3 + 4x2 contains two copies. I don't need three copies of "x", because neither polynomial contains three copies. I need only one copy of x + 2, because neither polynomial contains more than just the one copy. I need to account for the 2 from the second polynomial and the x + 3 from the first polynomial.
The LCM is 2x2(x + 2)(x + 3).
By the way, while you always multiply the factors together when you're finding the common denominators for regular fractions, you almost always want to leave the common denominators for polynomial fractions in factored form. That is, you'll need to multiply and simplify across the top, but don't multiply anything together across the bottom (in this case, don't multiply the x + 2 and the x + 3 to get x2 + 5x + 6). Remember: you'll still need to try to reduce the polynomial fraction when you're done simplifying across the top, so you'll need the bottom in factored form in the end, anyway.
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