uncover the Least usual Multiple (LCM) of 2 NumbersPractice renders Perfect

Find the Least typical Multiple (LCM) of two Numbers

One of the reasons we look at multiples and also primes is to use these approaches to discover the least typical multiple of 2 numbers. This will be useful when we add and subtract fractions with various denominators.

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Listing Multiples Method

A common multiple of two numbers is a number that is a lot of of both numbers. Intend we want to find common multiples that 10 and 25. We can list the first several multiples of every number. Then us look for multiples the are usual 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 view that (50) and also (100) show up in both lists. Lock are usual multiples the (10) and also (25). We would certainly find an ext common multiples if we ongoing the list of multiples because that each.

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






Prime components Method

Another way to uncover the least common multiple of 2 numbers is to usage their prime factors. We’ll use this technique to discover the LCM of (12) and also (18).

We start by finding the prime factorization of every number.

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

Then we compose each number as a product that primes, corresponding primes vertically once possible.

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

Now we lug down the primes in each column. The LCM is the product of this factors.

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Example (PageIndex7): lcm

Find the LCM that (50) and (100) making use of the prime components method.

Solution

Write the element factorization of each number. (50 = 2 cdot 5 cdot 5 qquad 100 = 2 cdot 2 cdot 5 cdot 5)
Write each number as a product that primes, corresponding 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 determinants to gain the LCM.

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LCM = 2 • 2 • 5 • 5

The LCM of 50 and also 100 is 100.


Exercise (PageIndex14)

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

price

(360)



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