Question 1209285: Find the LCM of 7, 10, 12, 15, 24, 75
Answer by math_tutor2020(3816)   (Show Source):
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Answer: 4200
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Explanation
I'll discuss 4 methods
Method 1
Find the prime factorization of each value.
If the original value is prime, then just say "1 times that number" even though 1 itself isn't prime.
7 = 1*7
10 = 2*5
12 = 2*2*3
15 = 3*5
24 = 2*2*2*3
75 = 3*5*5
The unique prime factors mentioned were: 2,3,5,7
For any given row,- 2 shows up at most three times to contribute 2^3 as part of the LCM.
- 3 shows up at most once to contribute 3^1
- 5 shows up at most twice to contribute 5^2
- 7 shows up at most once to contribute 7^1
The LCM is therefore (2^3)*(3^1)*(5^2)*(7^1) = 4200
Many online LCM calculators can be used to verify the answer.
If you are using Desmos then the command to type in is lcm(7, 10, 12, 15, 24, 75)
The "lcm" must be all lowercase. Do not use curly braces.
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Method 2
Let's say we only worried about finding the LCM of the set of two values {x,y}
We have a very useful formula that connects the LCM and GCF of two numbers.
LCM = x*y/GCF
where "LCM" refers to "LCM of x and y"; and "GCF" refers to "GCF of x and y".
That formula is going to be used repeatedly since we have more than two numbers in the original set.
Let's find the LCM of the subset {7,10} which are the first two items of the original set.
LCM = x*y/GCF
LCM = 7*10/1
LCM = 70
The LCM of {7,10} is 70
Replace "7,10" in the original set with "70"
{7, 10, 12, 15, 24, 75} shrinks to {70, 12, 15, 24, 75}
Then we do another LCM calculation on the first two items of this shrunken set.
LCM = x*y/GCF = 70*12/2 = 420
The LCM of the subset {70,12} is 420.
Replace "70,12" with "420"
{70, 12, 15, 24, 75} shrinks to {420, 15, 24, 75}
It's a somewhat slow process but we are steadily shrinking the set down.
Keep the process going until there's one number remaining.
You should arrive at 4200
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Method 3
I'll use the Cake method discussed on this page
Read that page first to see how the method works. Other names for it could be "ladder method", "factor box method", or "grid method". Perhaps there might be other names.
Here's the scratch work when using the Cake Method.
| (1) | 7 | 10 | 12 | 15 | 24 | 75 | | (2) | 7 | 5 | 6 | 15 | 12 | 75 | | (2) | 7 | 5 | 3 | 15 | 6 | 75 | | (3) | 7 | 5 | 1 | 5 | 2 | 25 | | (5) | 7 | 1 | 1 | 1 | 2 | 5 | | (7) | (1) | (1) | (1) | (2) | (5) |
p = multiply the left column of parenthesis values: 1*2*2*3*5 = 60
q = multiply the bottom row of parenthesis values: 7*1*1*1*2*5 = 70
LCM = p*q = 60*70 = 4200
Ignoring the (1), the first row is the original set of values. The (1) off to the left is optional but I find it helps with consistency.
The second row is where I divide even values by 2 (eg: 10/2 = 5). The (2) off to the left keeps track of what I divided the row by. If a value is not a multiple of 2, then leave it as is.
The third row is the same idea. We divide by another 2 since 6 and 12 have this as a common factor.
Once reaching "7, 5, 1, 5, 2, 25" it's clear that 2 is not a common factor of any pairing. So we then divide by 3, and then by 5, and so on. We only need to check up to the prime 7.
The bottom row of values in parenthesis is optional, but I find it helps with consistency.
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Method 4
The Division Table method is very similar to the Cake method.
Here's the scratch work
| (1) | 7 | 10 | 12 | 15 | 24 | 75 | | (2) | 7 | 5 | 6 | 15 | 12 | 75 | | (2) | 7 | 5 | 3 | 15 | 6 | 75 | | (2) | 7 | 5 | 3 | 15 | 3 | 75 | | (3) | 7 | 5 | 1 | 5 | 1 | 25 | | (5) | 7 | 1 | 1 | 1 | 1 | 5 | | (5) | 7 | 1 | 1 | 1 | 1 | 1 | | (7) | 1 | 1 | 1 | 1 | 1 | 1 |
Collect all the items in parenthesis to multiply together:
LCM = 1*2*2*2*3*5*5*7 = 4200
The Division Table method is discussed on this page. Each item in a parenthesis tells us what I divided some of the row items by. The goal of this method is to get nothing but 1's in the bottom row.
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Technically there's a 5th alternate route of listing the multiples of each item to see what they have in common.
However, this is NOT recommended since the LCM (4200) is very large.
You'll be listing a lot of multiples. Perhaps a spreadsheet is best suited for a task like this. Even then it would be tedious busy-work in my opinion.
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