244
CHAPTER 5
Number Theory
10
18
1
14
22
11
24
7
20
3
17
5
13
21
9
23
6
19
2
15
4
12
25
8
16
COLLABORATIVE INVESTIGATION
Investigating an Interesting Property of Number Squares
In the Extension at the end of this chapter, we looked at
magic squares. Now in this group activity we will investigate another property of squares of numbers.
Begin by dividing up the class into groups of three or
four students. Each student in the group should prepare a square of numbers like the one that follows:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Topics for Discussion
1. Each student should do the following individually:
Choose any number in the first row. Circle it, and
cross out all entries in the column below it. (For
example, if you circle 4, cross out 9, 14, 19, and
24.) Now circle any remaining number in the second row, and cross out all entries in the column
below it.
Repeat this procedure for the third and fourth
rows, and then circle the final remaining number
in the fifth row.
Now each student in the group should add the
circled numbers and compare his or her sum with
all others in the group. What do you notice?
2. How does the sum obtained in Exercise 1 compare
with the magic sum for an order 5 magic square?
3. Suppose Exercise 1 was done as shown here:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Notice that summing the circled entries is just
like summing 1 2 3 4 5, except that
3 is replaced by 3 5,
5 is replaced by 5 10,
1 is replaced by 1 15,
4 is replaced by 4 20 .
We can express this as
sum 1 2 3 4 5
5 10 15 20
15 50 65 .
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245
Chapter 5 Test
4. Explain why, whatever entries you choose to
circle in the various rows, the sum is always the
same.
5. Prepare a similar square of the natural numbers 1
through 36. Then repeat Exercise 1. Discuss your
results. How does the sum compare with the
magic sum for an order 6 magic square?
6. As a group, fill in the entries in this equation for
the 6 by 6 square.
sum (
(
)
)
7. As a group, predict the sum of the circled numbers in a 7 by 7 square by expressing it as follows.
(Do not actually construct the square.)
sum (
)(
)
How does the sum compare with the magic sum
for an order 7 magic square?
8. Each individual should now prepare another 5 by
5 square and repeat Exercise 1, except this time
start with a number in the first column and cross
out remaining numbers in rows. In your group,
discuss and explain what you observe.
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
In Exercises 1– 5, decide whether each statement is true
or false.
1. No two prime numbers differ by 1.
2. There are infinitely many prime numbers.
3. If a natural number is divisible by 9, then it must also
be divisible by 3.
4. If p and q are different primes, 1 is their greatest
common factor and pq is their least common multiple.
5. For all natural numbers n, 1 is a factor of n and n is
9. In your own words state the Fundamental Theorem
of Arithmetic.
10. Decide whether each number is perfect, deficient, or
abundant.
(a) 17
(b) 6
(c) 24
11. Which of the following statements is false?
A. There are no known odd perfect numbers.
B. Every even perfect number must end in 6 or 28.
C. Goldbach’s Conjecture for the number 8 is veri-
fied by the equation 8 7 1.
12. Give a pair of twin primes between 40 and 50.
a multiple of n.
6. Use divisibility tests to determine whether the number
13. Find the greatest common factor of 270 and 450.
331,153,470
14. Find the least common multiple of 24, 36, and 60.
is divisible by each of the following.
(b) 3
(c) 4
(a) 2
(e) 6
(f) 8
(d) 5
(h) 10
(i) 12
(g) 9
7. Decide whether each number is prime, composite, or
neither.
(a) 93
(b) 1
(c) 59
8. Give the prime factorization of 1440.
15. Day Off for Fast-food Workers Both Sherrie Firavich
and Della Daniel work at a fast-food outlet. Sherrie
has every sixth day off and Della has every fourth day
off. If they are both off on Wednesday of this week,
what will be the day of the week that they are next off
together?
16. The twenty-second Fibonacci number is 17,711 and
the twenty-third Fibonacci number is 28,657. What
is the twenty-fourth Fibonacci number?