Matemática Básica - Aula 12 - Máximo divisor comum - MDC

Introduction to Maximum Common Divisor

Overview of the Lesson

• The lesson focuses on the concept of the maximum common divisor (MDC), defined as the largest number that divides two or more numbers without leaving a remainder.

• Two methods will be utilized for calculations: Euclidean algorithm (method of successive divisions) and simultaneous decomposition into prime factors.

• The instructor emphasizes the importance of understanding MDC, also referred to as "greatest common divisor" (GCD).

Understanding Maximum Common Divisor

• MDC is described as the highest integer that can divide given numbers.

• Example provided with divisors of 18: 1, 2, 3, 6, 9, 18 and divisors of 30: 1, 2, 3, 5, 6, 10, 15, 30.

• Common divisors between both sets are identified: 1, 2, 3, 6, with the largest being six.

Euclidean Algorithm for Finding MDC

Introduction to Euclidean Algorithm

• The instructor introduces a simpler method called the Euclidean algorithm which does not require factorization.

Step-by-Step Example

• To find MDC between numbers like 160 and 64:

• Start by placing larger number (160) on left and smaller (64) on right.

• Calculate how many times the smaller fits into larger; here it fits twice (128).

• Remainder is calculated as 160 - 128 = 32.

Continuing with Euclidean Steps

• Repeat process using new pair: now compare smaller number (64) with remainder (32).

• Again calculate how many times it fits; it fits exactly twice this time resulting in zero remainder.

Conclusion from Example

• When remainder reaches zero in this process, last non-zero remainder is MDC. Here it's determined that MDC(160,64)=32.

Another Example Using Smaller Numbers

Finding MDC Between Small Numbers

• Another example involves finding MDC between numbers like twelve and twenty:

• Place larger number first; calculate how many times twelve fits into twenty yielding a remainder of eight.

Continuing Calculations

• Next step compares eight against twelve leading to another calculation yielding four as a new remainder.

Final Result for Small Numbers

• This process continues until reaching zero; thus concluding that MDC(12,20)=4.

Simultaneous Decomposition Method

Introduction to Prime Factorization Method

• A different approach involves simultaneous decomposition into prime factors which allows finding MDC among multiple numbers effectively.

Starting with an Example

Understanding the Maximum Common Divisor

Step-by-Step Calculation of GCD for 90 and 54

• The process begins by dividing 90 and 54 by 2, resulting in 45 and 27 respectively.

• Continuing with division by prime numbers, we divide both results by 3: 45 becomes 15 and 27 becomes 9.

• Further division shows that both numbers can be divided again by 3, yielding a final result of 1 for the last column.

• The factors that divided both rows simultaneously are marked; only the common factors (2 and one instance of three) are considered for calculating the GCD.

• The maximum common divisor is calculated as 2 times 3 times 3 = 18, which is confirmed as the GCD of both numbers.

Extending to Three Numbers: GCD of 70, 90, and 120

• A vertical line separates the three numbers; initial divisions show that all can be divided by two.

• Continuing with further divisions reveals that while some numbers can still be divided by two or three, others cannot.

• Only certain factors are marked based on their ability to divide all columns simultaneously; this leads to identifying relevant prime factors.

• The final calculation indicates that the maximum common divisor among these three numbers is 2 times 5 =10.

Properties of Maximum Common Divisor

Property One: Primes Always Yield a GCD of One

• When calculating the GCD between any two prime numbers, it will always equal one due to their indivisibility.

• Examples include pairs like (11,7), where no number greater than one divides them evenly.

Property Two: Divisor Implications

Maximizing Common Divisors: Understanding GCD

Introduction to Maximum Common Divisor (MDC)

• The maximum common divisor (MDC) between two numbers, such as 4 and 20, is simply the smaller number if it divides the larger one. In this case, MDC(4, 20) = 4.

Properties of Maximum Common Divisor

• When calculating the MDC of two numbers like 8 and 12, the result is 4. This illustrates a property similar to that of the least common multiple (LCM).

• Doubling or tripling both numbers does not change their MDC. For example, doubling gives us new pairs (16 and 24), but MDC remains consistent at 8 when calculated from doubled results.

Division Property in GCD Calculation

• If you divide both numbers by a common factor (e.g., dividing 16 by 4 and 24 by 4), you can find the new MDC without recalculating from scratch; here, MDC(4,6) = 2.

Relationship Between LCM and GCD

• A crucial relationship exists between LCM and GCD: LCM(A,B) * GCD(A,B) = A * B. This means you can multiply the original numbers directly for quick calculations.

• For instance, using numbers like 12 and 18: instead of calculating LCM(12,18)=36 and GCD(12,18)=6 separately to get their product as 216, you can simply multiply them directly.

Application in Problem Solving

• The concepts discussed are frequently tested in exams like ENEM. Understanding how to apply these properties is essential for solving related problems efficiently.

Practical Example: Calculating Squares on a Wall

Problem Setup

• A problem involves tiling a rectangular wall with squares while ensuring no gaps or cuts occur. The dimensions given are width = 8.8 m and height = 5.5 m.

Finding Minimum Square Size

• To determine the minimum number of squares needed without cutting any pieces, we need to find the largest square size that fits both dimensions perfectly.

Example Calculation with Dimensions

• Using an example where one dimension is set at length =10m and height =8m helps illustrate finding suitable square sizes; here a side length of '2' allows fitting perfectly into both dimensions.

Importance of Divisibility

• The chosen square size must be a divisor of both wall dimensions; thus '2' works while '4' does not fit evenly into '10', leading to leftover space.

Advanced Calculation Techniques

Converting Measurements for Accuracy

• To handle decimal measurements effectively (like converting meters to centimeters), it's necessary to convert all values before performing calculations—880 cm for width and 550 cm for height.

Factorization Approach

• Instead of working with large values directly (880 &550), factorization simplifies finding the MDC through smaller integers (88 &55).

Final Result Interpretation

Understanding the Greatest Common Divisor in Practical Applications

Finding the GCD for Larger Numbers

• The discussion begins with a focus on finding the greatest common divisor (GCD) between 880 and 550, rather than between their smaller counterparts, 88 and 55. This emphasizes the importance of scaling numbers appropriately for practical applications.

• After multiplying both numbers by 10 to obtain 880 and 550, it is established that the GCD is now relevant to these larger values. The calculated GCD is confirmed as 110.

Application of GCD in Area Calculation

• A real-world scenario is presented involving a wall measuring 880 cm in length and 550 cm in height. The question posed is how many squares with sides of length 110 cm can fit within this area.

• It’s determined that five squares can fit vertically (in height) and eight squares horizontally (in length), leading to a total of 5 times 8 = 40 squares fitting into the wall space.

Extending Concepts to Three Dimensions

• Transitioning from two-dimensional calculations, the discussion shifts to three-dimensional shapes, specifically focusing on filling a rectangular parallelepiped completely with cubes.

• To achieve this, participants are tasked with finding the GCD among three dimensions: 8, 20, and 36. The factorization process reveals that the GCD is determined to be 2 times 2 = 4.

Fitting Cubes into Dimensions

• With a cube side length of four units established from the previous calculation, it’s noted that two cubes will fit vertically within an eight-unit height.