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Find the Least common Multiple (LCM) of 2 Numbers

One that the reasons we look at multiples and primes is to use these methods to uncover the least usual multiple of 2 numbers. This will certainly be beneficial when we include and subtract fountain with various denominators.

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Listing Multiples Method

A usual multiple of 2 numbers is a number the is a lot of of both numbers. Mean we want to find typical multiples of 10 and also 25. We deserve to list the first several multiples of every number. Then we look because that multiples that are common to both lists—these are the typical multiples.

<eginsplit 10 & colon ; 10, 20, 30, 40, extbf50, 60, 70, 80, 90, extbf100, 110, ldots \ 25 & colon ; 25, extbf50, 75, extbf100, 125, ldots endsplit onumber >

We watch that (50) and (100) appear in both lists. Castle are usual multiples the (10) and also (25). We would find more common multiples if we continued the perform of multiples for each.

The smallest number the is a multiple of two numbers is called the least usual multiple (LCM). For this reason the least LCM of (10) and also (25) is (50).






Prime factors Method

Another means to uncover the least usual multiple of two numbers is to usage their element factors. We’ll usage this technique to discover the LCM the (12) and (18).

We begin by finding the element factorization of each number.

<12 = 2 cdot 2 cdot 3 qquad qquad 18 = 2 cdot 3 cdot 3 onumber>

Then we create each number as a product that primes, matching primes vertically once possible.

<eginsplit 12 & = 2 cdot 2 cdot 3 \ 18 & = 2 cdot quad ; 3 cdot 3 endsplit onumber >

Now we carry down the primes in every column. The LCM is the product of this factors.

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Example (PageIndex7): lcm

Find the LCM that (50) and (100) utilizing the prime determinants method.

Solution

Write the element factorization of every number. (50 = 2 cdot 5 cdot 5 qquad 100 = 2 cdot 2 cdot 5 cdot 5)
Write each number together a product that primes, corresponding primes vertically once possible. (eginsplit 50 & = quad ; 2 cdot 5 cdot 5 \ 100 & = 2 cdot 2 cdot 5 cdot 5 endsplit)
Bring down the primes in each column.
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Multiply the determinants to gain the LCM.

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LCM = 2 • 2 • 5 • 5

The LCM that 50 and also 100 is 100.


Exercise (PageIndex14)

Find the LCM using the prime determinants method: (60, 72)

price

(360)



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Grocery shopping warm dogs are sold in packages that ten, however hot dog buns come in packs of eight. What is the smallest number of hot dogs and also buns that have the right to be purchased if you want to have the same number of hot dogs and buns? (Hint: the is the LCM!) Grocery shopping file plates are offered in packages of 12 and party cup come in package of 8. What is the smallest number of plates and also cups you deserve to purchase if you want to have actually the same variety of each? (Hint: it is the LCM!)