CHAPTER 5 - PROBLEMS FOR WRITING AND DISCUSSION
1. Draw a Venn Diagram with three circles and place the factors of 24 in one circle, the
factors of 36 in the second circle and 57 in the third circle. Pay close attention to
numbers that belong to two or three circles simultaneously. Use the diagram to devise
a method for finding the GCF and LCM of 24, 36, and 57. Describe and justify our
method.
2. Write out all the numbers from 1 to 100 in a six-column array, beginning as follows:
1 2 3 4 5 6
7 8 9 10 11 12
Proceed as in the Sieve of Eratosthenes given in Section 5.1. That is, circle every
prime number and then draw a line through all the multiples of that number, starting
with 2, then 3, etc. What do you notice about the multiples of 5? of 7? Once you have
crossed out all the multiples of 7, have you found all the primes from 1 to 100? How
do you know?
3. In mathematics, what is the difference between a "conjecture" and a "theorem"?
(Hint: See the Focus On at the beginning of chapter 5.) Was your answer to the
preceding problem a conjecture or a theorem? If it was a conjecture, how could you
turn it into a theorem?
4. It is true that if 2 | 36 and 9 | 36, then 18 | 36, since 18 = 2 × 9. So wouldn't it also be
true that since 4 | 36 and 6 | 36, then 24 | 36 since 24 = 4 × 6? Why or why not?
5. Although most people associate Euclid's name most closely with geometry, he is also
given credit for three of the ideas about number theory that are contained in this
section. He was the first person to prove there are an infinite number of prime
numbers. State the other two ideas in this section for which we give Euclid credit, and
show an example of each.
6. When computing the least common multiple of several pairs of numbers, a student
notices a pattern:
LCM(4, 7) = 28
LCM(8, 9) = 72
LCM(5, 11) = 55
He says, “I discovered a shortcut for finding the least common multiple of two
numbers. You just multiply the numbers together.”
a. When will this student’s shortcut work?
b. If the student tries to find the LCM(8, 10) using his shortcut, he will get 80.
Using the prime factorization method presented in this chapter, what answer
should the student get? Show the process.
c. After using prime factorization to find the LCM(8, 10) = 40, the student says, “If
you divide 80 by 2, you get 40.” What relationship has the student noticed; that
is, in the context of what you have learned in this chapter, how is 2 related to 8
and 10?
d. How would the student revise his original shortcut so that it applies to finding the
LCM of any pair of counting numbers? Choose another pair of numbers and try
the new shortcut.