Problem 38 Find the LCM. $$18,54,63$$... [FREE SOLUTION]

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Chapter 2: Problem 38

Find the LCM. $$18,54,63$$

Short Answer

Expert verified

The least common multiple (LCM) of 18, 54, and 63 is $2\times 3^3\times 7= 378$

Step by step solution

Prime Factorization of Each Number

Find the prime factorization of each number: $18=2\times 3^2$, $54=2\times 3^3$, $63=3^2\times 7$

Identify the Highest Power of Each Prime Factor

From the three numbers, identify the highest power of each prime factor: $2^1$, $3^3$, $7^1$

Find the Least Common Multiple

The LCM is found by multiplying the highest power of all the prime numbers: $2\times 3^3\times 7$. This should give you the least common multiple of 18, 54, and 63.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factorization

Prime factorization is a method of breaking down numbers into their smallest building blocks, called prime factors. A prime number is a number that has no divisors other than 1 and itself. To perform prime factorization, you keep dividing a number by its smallest prime divisor until you can't go any further.

For instance, let's take the number 18.

Divide 18 by 2, the smallest prime number. You get 9.
Then, divide 9 by 3, another prime number, to get 3.
Finally, divide by 3 once more to get 1.

Therefore, the prime factorization of 18 is expressed as $2 \times 3^2$. There it is! We've found the prime factors of 18 by division. Breaking down a number this way helps uncover its prime factors, like revealing what ingredients make up a delicious recipe.

Multiplying Factors

Multiplying factors is the process we use to put numbers or elements together to get a product. When it comes to finding the Least Common Multiple (LCM), we take the prime factors from each number and multiply them together, but with a twist!

We multiply only the highest powers of each prime factor found in the numbers being considered.

For example, say we are working with the numbers 18, 54, and 63. We first carry out the prime factorization:

- 18: $2 \times 3^2$- 54: $2 \times 3^3$- 63: $3^2 \times 7$
Notice how we pick the highest power of each prime number: 2 from 18, 3 from 54, and 7 from 63. Finding the LCM involves bringing those biggest power teammates together and letting them multiply to reveal the LCM.

Highest Power of Prime Factors

Finding the highest power of prime factors means choosing the largest exponent for each prime factor from a set of numbers during their prime factorization. This is a crucial step in calculating the Least Common Multiple (LCM).

Let's break it down a bit more.

From the factorization $18 = 2 \times 3^2$ we note that 2 appears with power 1 and 3 with power 2.
From $54 = 2 \times 3^3$, similarly, 2 also appears but with power 1 and 3 with power 3.
Finally, from $63 = 3^2 \times 7$, 3 appears with power 2, and 7 emerges with power 1.

What we did in the exercise was to choose the highest exponent for each distinct prime factor across those numbers. For LCM calculation: - 2 shows up as $2^1$ - 3 comes in strong with $3^3$ - 7 steps in with $7^1$By picking and multiplying these highest powers together, we assure that the LCM for all numbers in question covers all prime factors necessary to be divisible by each of the original numbers.

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91影视

Find the LCM. $$18,54,63$$

Short Answer

Step by step solution

Prime Factorization of Each Number

Identify the Highest Power of Each Prime Factor

Find the Least Common Multiple

Key Concepts

Prime Factorization

Multiplying Factors

Highest Power of Prime Factors

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