Lowest Common Multiple Of 16 And 28

Finding the Lowest Common Multiple (LCM) of 16 and 28

This article will guide you through finding the lowest common multiple (LCM) of 16 and 28. Understanding LCM is crucial in various mathematical contexts, from simplifying fractions to solving problems involving cyclical events. We'll explore two primary methods: the prime factorization method and the least common multiple formula method. Learn how to calculate the LCM efficiently and confidently.

What is the Lowest Common Multiple (LCM)?

The lowest common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by all of the integers. In simpler terms, it's the smallest number that both numbers can divide into evenly. For example, the LCM of 2 and 3 is 6, because 6 is the smallest number divisible by both 2 and 3.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors. The prime factors are the prime numbers that multiply together to make the original number.

1. Find the prime factorization of 16:

   16 = 2 x 2 x 2 x 2 = 2⁴

2. Find the prime factorization of 28:

   28 = 2 x 2 x 7 = 2² x 7

3. Identify the highest power of each prime factor:

   The prime factors are 2 and 7. The highest power of 2 is 2⁴ (from the factorization of 16), and the highest power of 7 is 7¹ (from the factorization of 28).

4. Multiply the highest powers together:

   LCM(16, 28) = 2⁴ x 7 = 16 x 7 = 112

Therefore, the lowest common multiple of 16 and 28 is 112.

Method 2: Using the Formula (LCM and GCD)

This method utilizes the relationship between the LCM and the greatest common divisor (GCD) of two numbers. The formula states:

LCM(a, b) = (|a x b|) / GCD(a, b)

where:

• a and b are the two numbers.
• GCD(a, b) is the greatest common divisor of a and b.

1. Find the GCD of 16 and 28:

   We can use the Euclidean algorithm to find the GCD.

   • 28 = 16 x 1 + 12
   • 16 = 12 x 1 + 4
   • 12 = 4 x 3 + 0

   The last non-zero remainder is 4, so GCD(16, 28) = 4.

2. Apply the formula:

   LCM(16, 28) = (16 x 28) / 4 = 448 / 4 = 112

Again, the lowest common multiple of 16 and 28 is 112.

Conclusion:

Both methods effectively determine the LCM of 16 and 28, yielding the same result: 112. Choosing the best method depends on your preference and the complexity of the numbers involved. The prime factorization method is generally easier for smaller numbers, while the formula method can be more efficient for larger numbers, especially when a GCD calculator is available. Understanding these methods empowers you to solve various mathematical problems involving multiples and divisors with confidence.