least common multiple of 8 and 4

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31-03-2025

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The least common multiple (LCM) is the smallest positive integer that is divisible by both numbers without leaving a remainder. Let's find the LCM of 8 and 4. We'll explore a few different methods to solve this.

Method 1: Listing Multiples

The simplest method is to list the multiples of each number until you find the smallest multiple they share.

• Multiples of 8: 8, 16, 24, 32, 40, ...
• Multiples of 4: 4, 8, 12, 16, 20, 24, ...

Notice that 8 is a multiple of both 8 and 4. Also, 16 and 24 appear in both lists. The smallest common multiple is 8. Therefore, the LCM of 8 and 4 is 8.

Method 2: Prime Factorization

This method is particularly useful for larger numbers. We break down each number into its prime factors.

• Prime factorization of 8: 2 x 2 x 2 = 2³
• Prime factorization of 4: 2 x 2 = 2²

To find the LCM, we take the highest power of each prime factor present in either factorization. In this case, the only prime factor is 2, and the highest power is 2³ (from the factorization of 8).

Therefore, LCM(8, 4) = 2³ = 8.

Method 3: Using the Formula (for two numbers)

There's a handy formula to calculate the LCM of two numbers (a and b) if you know their greatest common divisor (GCD):

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

First, find the GCD (greatest common divisor) of 8 and 4. The greatest common divisor is the largest number that divides both 8 and 4 without leaving a remainder. In this case, the GCD(8, 4) = 4.

Now, apply the formula:

LCM(8, 4) = (8 * 4) / 4 = 32 / 4 = 8

Understanding the Results

The LCM of 8 and 4 is 8. This means that 8 is the smallest number that is divisible by both 8 and 4. Any other common multiple (like 16, 24, etc.) will be a multiple of 8.

Applications of LCM

Finding the least common multiple has practical applications in various areas, including:

• Scheduling: Determining when events will occur simultaneously. For example, if one event happens every 8 days and another every 4 days, they will coincide every 8 days.
• Fractions: Finding a common denominator when adding or subtracting fractions.
• Measurement: Converting between different units of measurement.

This comprehensive guide provides you with various methods to calculate the least common multiple, ensuring you can tackle similar problems with ease. Remember to choose the method that best suits the numbers involved.