discover the Least common Multiple (LCM) of 2 NumbersPractice renders Perfect

Find the Least typical Multiple (LCM) of two Numbers

One the the reasons we look in ~ multiples and primes is to usage these techniques to uncover the least usual multiple of two numbers. This will certainly be valuable when we include and subtract fountain with different 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 common multiples that 10 and also 25. We have the right to list the very first several multiples of each number. Then us look because that multiples the are common to both lists—these room the usual 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 check out that (50) and also (100) appear in both lists. Castle are common multiples the (10) and also (25). We would certainly find more common multiples if we continued the list of multiples because that each.

The smallest number that is a lot of of 2 numbers is called the least usual multiple (LCM). For this reason the the very least LCM the (10) and (25) is (50).






Prime components Method

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

We start 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 compose each number as a product the primes, matching primes vertically as soon as possible.

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

Now we lug down the primes in every column. The LCM is the product of these factors.

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Example (PageIndex7): lcm

Find the LCM that (50) and also (100) using the prime factors 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 every number together a product that primes, equivalent 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 components to obtain the LCM.

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LCM = 2 • 2 • 5 • 5

The LCM that 50 and 100 is 100.


Exercise (PageIndex14)

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

answer

(360)



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Grocery shopping hot dogs are offered in packages of ten, yet hot dog buns come in packs of eight. What is the smallest variety 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: it is the LCM!) Grocery shopping document plates are sold in packages of 12 and also party cups come in package of 8. What is the smallest number of plates and cups you deserve to purchase if you desire to have the same number of each? (Hint: it is the LCM!)