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Challenge The GCF and the LCM
jOb ective Think analytically about common factors and common multiples.
Challenge The GCF and the LCM 1. Textbook Instructions
1 . Think about the following problems and answer them. 1. This challenge activity calls on students to
think analytically and contextually about
a. Is it possible for the GCF of three numbers to be equal to one of the common factors and multiples.
numbers? Use an example to explain.
1a. Can the greatest common factor of
The GCF of three numbers is equal to one of the numbers when all the three numbers be one of those numbers?
numbers are multiples of that number. Yes. (In fact, 1d in Activity 2 was an example,
For example, if 7 is the GCF, the three numbers can be 7, 21, and 49. with values 39, 13, and 26, the GCF being
13.)
b. Why do you think mathematicians don’t try to find the least common
factor or the greatest common multiple? Explain. 1b. Why do mathematicians not examine
the least common factor or greatest
The least common factor of all numbers is 1. common multiple? The first of these cases is
Since it is always the same, there is no need to find it. Also, the greatest trivial; the least common factor of every
common multiple cannot be found because numbers go up to infinity. integer is 1. The second case has no
numerical solution, since common multiples
can be extended infinitely.
2. Which Is the Greatest? 17
2. Build Understanding
This challenge offers an opportunity to introduce students to the general concept of trivial cases,
mathematical relationships that are too simple to yield useful insights.
The least common factor of integers is such an example, because it is always 1. Not only is the answer
“trivially” easy, it is also not helpful.
Example: If we use 1 as the chosen common factor in constructing a ladder diagram to determine the
greatest common factor, the second step of the ladder simply returns the original numbers, getting us
nowhere.
1 15 8 7
15 8 7
6A Unit 02 025
jOb ective Think analytically about common factors and common multiples.
Challenge The GCF and the LCM 1. Textbook Instructions
1 . Think about the following problems and answer them. 1. This challenge activity calls on students to
think analytically and contextually about
a. Is it possible for the GCF of three numbers to be equal to one of the common factors and multiples.
numbers? Use an example to explain.
1a. Can the greatest common factor of
The GCF of three numbers is equal to one of the numbers when all the three numbers be one of those numbers?
numbers are multiples of that number. Yes. (In fact, 1d in Activity 2 was an example,
For example, if 7 is the GCF, the three numbers can be 7, 21, and 49. with values 39, 13, and 26, the GCF being
13.)
b. Why do you think mathematicians don’t try to find the least common
factor or the greatest common multiple? Explain. 1b. Why do mathematicians not examine
the least common factor or greatest
The least common factor of all numbers is 1. common multiple? The first of these cases is
Since it is always the same, there is no need to find it. Also, the greatest trivial; the least common factor of every
common multiple cannot be found because numbers go up to infinity. integer is 1. The second case has no
numerical solution, since common multiples
can be extended infinitely.
2. Which Is the Greatest? 17
2. Build Understanding
This challenge offers an opportunity to introduce students to the general concept of trivial cases,
mathematical relationships that are too simple to yield useful insights.
The least common factor of integers is such an example, because it is always 1. Not only is the answer
“trivially” easy, it is also not helpful.
Example: If we use 1 as the chosen common factor in constructing a ladder diagram to determine the
greatest common factor, the second step of the ladder simply returns the original numbers, getting us
nowhere.
1 15 8 7
15 8 7
6A Unit 02 025
