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Find the Least typical Multiple (LCM) of 2 Numbers

One the the factors we look at multiples and also primes is to use these methods to discover the least common multiple of two numbers. This will certainly be helpful when we include and subtract fountain with different denominators.

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Listing Multiples Method

A usual multiple of two numbers is a number that is a many of both numbers. Expect we want to find typical multiples of 10 and also 25. We can list the first several multiples of each number. Then us look for 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 (100) show up in both lists. They are usual multiples of (10) and (25). We would find more common multiples if we continued the list of multiples because that each.

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






Prime factors Method

Another method to discover the least typical multiple of 2 numbers is to usage their element factors. We’ll usage this an approach to find the LCM of (12) and also (18).

We start by detect the prime factorization of each number.

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

Then we compose each number together a product that primes, equivalent primes vertically when possible.

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

Now we bring down the primes in each column. The LCM is the product of these factors.

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Example (PageIndex7): lcm

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

Solution

Write the prime 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, matching primes vertically when possible. (eginsplit 50 & = quad ; 2 cdot 5 cdot 5 \ 100 & = 2 cdot 2 cdot 5 cdot 5 endsplit)
Bring down the primes in every column.
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Multiply the factors to gain the LCM.

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LCM = 2 • 2 • 5 • 5

The LCM that 50 and 100 is 100.


Exercise (PageIndex14)

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

prize

(360)



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