Math Ed 305 Defining Common Divisors and Multiples. 1. From a partitive perspective, to say that X is a divisor of Y is to say that:

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1 Defining Common Divisors and Multiples Part A. 1. From a partitive perspective, to say that X is a divisor of Y is to say that: 2. From a measurement perspective, to say that X is a divisor of Y is to say that: 3. From a partitive perspective, to say that M is a multiple of N is to say that: 4. From a measurement perspective, to say that M is a multiple of N is to say that:

2 Part B. 5. From a partitive perspective, to say that X is a common divisor of Y and Z is to say that: 6. From a measurement perspective, to say that X is a common divisor of Y and Z is to say that: 7. From a partitive perspective, to say that M is a common multiple of N and P is to say that: 8. From a measurement perspective, to say that M is a common multiple of N and P is to say that:

3 Part C. 9. From a partitive perspective, to say that X is the greatest common divisor of Y and Z is to say that: 10. From a measurement perspective, to say that X is the greatest common divisor of Y and Z is to say that: 11. From a partitive perspective, to say that M is the least common multiple of N and P is to say that: 12. From a measurement perspective, to say that M is the least common multiple of N and P is to say that:

4 Using Squares 1. A student claimed that the LCM of X and Y will be the side of the smallest square you can build using copies of an X by Y rectangle. Is this true? Explain why or why not. Does your explanation use a measurement or partitive viewpoint?

5 2. Another student claims that the GCD of X and Y is the largest square that you can use to tile an X by Y rectangle. Why or why not? Does your explanation use a measurement or partitive viewpoint?

6 GCD and LCM with Groups of Groups of Groups We discussed last time how 36 tiles could be divided into 2 groups of 3 groups of 2 groups of 3 or 3 groups of 2 groups of 2 groups of 3 and so on. In other words, 36 = Similarly, 24 = Using the idea of groups of groups of groups..., find the GCD of 36 and 24, and explain why it is the GCD using grouping language. Does your explanation use a measurement or partitive viewpoint?

7 2. Using once again the fact that 36 = and 24 = and the idea of groups of groups of groups..., find the LCM of 36 and 24, and explain why it is the LCM using grouping language. Does your explanation use a measurement or partitive viewpoint?

8 Using Prime Factorizations Consider the two numbers A and B already factored into a product of prime numbers: A = 2 x 2 x 2 x 3 x 7 B = 2 x 3 x 3 x 5 Fill in the following chart with yes and no as appropriate: Quantity Factor of A? Factor of B? Multiple of A? Multiple of B? 2 x 3 2 x 5 2 x 3 x 5 x 7 2 x 3 x 3 x 5 x 7 2 x 2 x 2 x 3 x 7 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 7 A) How do the prime factors help find these numbers or answer the above questions? When is a number a factor of A? When is it a multiple of A? B) Proceed to find the least common multiple (LCM) and greatest common divisor (GCD) of A and B using the prime factorizations.

9 Homework #6 Homework on GCD and LCM 1. Find the GCD of: a. 144 and 56 b. 72 and 180 c. 360 and Find the LCM of : a. 16 and 18 b. 12 and 32 c. 48 and A band of pirates divide 185 gold pieces and 148 pieces of silver. They were absolutely fair, and divided both evenly. How many pirates could there have been? Explain in terms of grouping. Did you use LCM or GCD? Was it a partitive or measurement viewpoint? 3. Jack and Jill dive into a pool at the same time and begin swimming laps. Jack swims at the steady rate of 1 length every 84 seconds. Jill swims at the rate of 1 length every 78 seconds. How many lengths will Jill have swum at the moment when she and Jack first reach the same end simultaneously? Describe your answer in terms of the GCD or LCM as appropriate. Did you use a partitive or measurement viewpoint? 4. Two neon signs are turned on at the same time. One blinks every 4 seconds; the other blinks every 6 seconds. How many times per minute do they blink together? How did you use the LCM or GCD? 5. A certain warehouse contains 720 cartons of books. Can these cartons be placed in stacks of four cartons high so that each stack contains an equal number of cartons? Can they be placed in stacks five cartons high? six cartons high? seven cartons high? What is the greatest number of cartons that can be stacked so that each stack contains the same number of cartons and no stack is over 25 cartons high?