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Math

QuestionFind the least common multiple of 198 and 54. What is it?

Studdy Solution

STEP 1

Assumptions1. We are asked to find the least common multiple (LCM) of198 and54. . The LCM of two numbers is the smallest number that is a multiple of both numbers.

STEP 2

First, we need to find the prime factorization of each number.For198, we can start by dividing it by the smallest prime number, which is2.
198=2×99198 =2 \times99

STEP 3

We continue the prime factorization for99. The smallest prime number that divides99 is3.
99=3×3399 =3 \times33

STEP 4

We continue the prime factorization for33. The smallest prime number that divides33 is3.
33=3×1133 =3 \times11

STEP 5

So, the prime factorization of198 is198=2×3×3×11198 =2 \times3 \times3 \times11

STEP 6

Now, we find the prime factorization of54. We start by dividing it by the smallest prime number, which is2.
54=2×2754 =2 \times27

STEP 7

We continue the prime factorization for27. The smallest prime number that divides27 is3.
27=3×927 =3 \times9

STEP 8

We continue the prime factorization for. The smallest prime number that divides is3.
=3×3 =3 \times3

STEP 9

So, the prime factorization of54 is54=2×3×3×354 =2 \times3 \times3 \times3

STEP 10

To find the LCM, we take the highest power of each prime number from the prime factorizations of both numbers.
The highest power of2 is (from both198 and54), the highest power of3 is3 (from54), and the highest power of is (from198).
So, the LCM isLCM =2^ \times3^3 \times^

STEP 11

Calculate the LCM.
LCM=×27×11=594LCM = \times27 \times11 =594The least common multiple of198 and54 is594.

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