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

Find the Least typical Multiple (LCM) of 2 Numbers

One that the reasons we look at multiples and also primes is to use these methods to uncover the least typical multiple of 2 numbers. This will certainly be helpful when we include and subtract fractions with different denominators.

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Listing Multiples Method

A common multiple of 2 numbers is a number the is a lot of of both numbers. Intend we desire to find usual multiples that 10 and 25. We have the right to list the an initial several multiples of each number. Then us look for multiples that are usual to both lists—these are the common multiples.

\<\beginsplit 10 & \colon \; 10, 20, 30, 40, \textbf50, 60, 70, 80, 90, \textbf100, 110, \ldots \\ 25 & \colon \; 25, \textbf50, 75, \textbf100, 125, \ldots \endsplit \nonumber \>

We see that \(50\) and also \(100\) appear in both lists. Castle are usual multiples the \(10\) and also \(25\). We would find much more common multiples if we continued the list of multiples because that each.

The smallest number the is a multiple of 2 numbers is referred to as the least typical multiple (LCM). Therefore the least LCM of \(10\) and also \(25\) is \(50\).






Prime determinants Method

Another method to discover the least usual multiple of two numbers is to use their element factors. We’ll usage this method to discover the LCM the \(12\) 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 \nonumber\>

Then we compose each number together a product the primes, corresponding primes vertically once possible.

\<\beginsplit 12 & = 2 \cdot 2 \cdot 3 \\ 18 & = 2 \cdot \quad \; 3 \cdot 3 \endsplit \nonumber \>

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 of \(50\) and \(100\) using 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 of primes, corresponding primes vertically once possible. \(\beginsplit 50 & = \quad \; 2 \cdot 5 \cdot 5 \\ 100 & = 2 \cdot 2 \cdot 5 \cdot 5 \endsplit\)
Bring under the primes in every column.
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Multiply the determinants to acquire 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 utilizing the prime factors method: \(60, 72\)

price

\(360\)



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