Question

Find the least common multiple of the given numbers. \(20,16,\) and 12

Short answer

The LCM is 240.

Step-by-step solution

- Find Prime Factors

First, determine the prime factors for each number. For 20: 20 = 2^2 * 5For 16: 16 = 2^4For 12: 12 = 2^2 * 3

- Identify Highest Powers

List all prime numbers appearing in the factorization of the given numbers with their highest powers:2 appears as 2^4 (from 16)3 appears as 3 (from 12)5 appears as 5 (from 20)

- Multiply the Highest Powers

The least common multiple (LCM) is found by multiplying these highest powers together:LCM = 2^4 * 3^1 * 5^1LCM = 16 * 3 * 5LCM = 48 * 5LCM = 240

Key concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its prime number components.
Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
To find the prime factors of a number, start by dividing the number by the smallest prime (usually 2) and continue dividing the quotient by prime numbers until you end up with 1.
For example, to find the prime factors of the number 16, divide by 2:
\(16 = 2 \times 8\)
Then divide 8 by 2:
\(8 = 2 \times 4\)
Continue dividing by 2:
\(4 = 2 \times 2\)
Thus, the prime factorization of 16 is \(2^4\).

For the numbers in our problem:
20 is \(2^2 \times 5\)
16 is \(2^4\)
12 is \(2^2 \times 3\).

Highest Powers

After finding the prime factors, the next step is to determine the highest power of each prime number.
The highest power is simply the largest exponent of each prime number found when factoring all the numbers involved.
In our example:
The highest power of 2 is \(2^4\) (appearing in 16)
The highest power of 3 is 3 (appearing in 12)
The highest power of 5 is 5 (appearing in 20)

We look at each prime number and note the largest exponent to ensure these primes can cover all factors of the original numbers.

Multiplication of Factors

Once we have identified the prime factors and their highest powers, we find the least common multiple (LCM) by multiplying these factors together.
This will include all necessary prime components to cover all the original numbers.
For our example, multiply the prime factors with the highest powers:
\(LCM = 2^4 \times 3^1 \times 5^1\)
To calculate:
First, calculate \(2^4 = 16\)
Next, multiply 16 by 3:
\(16 \times 3 = 48\)
Then, multiply 48 by 5:
\(48 \times 5 = 240\)
Therefore, the least common multiple is 240.
This method ensures that all numbers can divide into the LCM without any remainder.