Calculating NWD (Greatest Common Divisor) and NWW (Least Common Multiple)

This page provides a comprehensive guide on how to calculate the Największy wspólny dzielnik (NWD) and Najmniejsza wspólna wielokrotność (NWW) using prime factorization methods. These concepts are fundamental in mathematics and have various applications in problem-solving.

Calculating NWD (Greatest Common Divisor)

To calculate the NWD, we follow these steps:

1. Decompose the given numbers into their prime factors.
2. Identify the common factors in both decompositions.
3. Multiply these common factors to obtain the NWD.

Example: Let's calculate the NWD of 840 and 540.

The prime factorization process is shown for both numbers:

840 = 2 × 2 × 2 × 3 × 5 × 7 540 = 2 × 2 × 3 × 3 × 5

The common factors are 2, 2, 3, and 5.

Highlight: NWD (840, 540) = 2 × 2 × 3 × 5 = 60

Calculating NWW (Least Common Multiple)

To calculate the NWW, we follow these steps:

1. Decompose the given numbers into their prime factors.
2. Include all factors from both decompositions, using the highest power of each prime factor that appears.
3. Multiply these factors to obtain the NWW.

Example: Let's calculate the NWW of 150 and 315.

The prime factorization process is shown for both numbers:

150 = 2 × 3 × 5 × 5 315 = 3 × 3 × 5 × 7

To find the NWW, we include all factors, using the highest power of each:

Highlight: NWW (150, 315) = 2 × 3 × 3 × 5 × 5 × 7 = 3150

This method ensures that we find the smallest number that is divisible by both 150 and 315.

Vocabulary:

• NWD: Największy wspólny dzielnik (Greatest Common Divisor)
• NWW: Najmniejsza wspólna wielokrotność (Least Common Multiple)
• Czynniki pierwsze: Prime factors

These techniques for calculating NWD and NWW are essential tools in mathematics, particularly useful in simplifying fractions, solving equations, and various other mathematical applications.