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Mathematical Background Notes
As preparation for the study of fractions, it is important for the student to
master some basic divisibility properties of whole numbers. This interesting topic leads to concepts such as prime and composite numbers, greatest common divisor, and least common multiple.
The fact that both methods lead to the same list of primes is a consequence of the Fundamental Theorem of Arithmetic. This important theorem asserts that, aside from the order of the factors, every whole
number greater than one has a unique prime factorization.
Lesson 2.1
The search for factors can be aided by divisibility rules for determining
whether special primes, such as 2, 3, 5, 7, or 11 are divisors of a given
number. Students will readily accept that a number is even if and only if
the last digit is even. However, as a basis for establishing less obvious
divisibility rules, it is useful to give a careful explanation of this result.
DIVISORS Division problems of the form b a are an important part of
the arithmetic of whole numbers. For a 0, it is possible to express the
solution of every such problem in the form
b q a r where q and r are whole numbers and 0 ∞ r ∞ a.
For example, the problem 73 6 has quotient 12 and remainder 1. This
corresponds to writing
73 12 6 1.
In case r 0, it is possible to write b q a. Here we say that b is
divisible by a and a is called a divisor or factor of b. Given a whole
number b, it is of interest to list all divisors of b in increasing order.
Divisors of 12: 1, 2, 3, 4, 6, 12
Divisors of 28: 1, 2, 4, 7, 14, 28
Divisors of 31: 1, 31
Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Whole numbers that are greater than 1 and have exactly two divisors, 1
and the number itself, are called prime numbers. Whole numbers with
more than two divisors are called composite numbers.
An important property of composite numbers is that they can be written
as the product of numbers smaller than themselves. For example,
12 2 6 3 4 2 2 3
25 5 5
32 2 16 4 8 4 4 2 2 2 8
222422222
When the number has been written as the product of prime factors, the
process of breaking it down into smaller factors terminates (we do not
include multiplication by 1). Writing a number as the product of primes is
an important skill to be learned in Lesson 2.1.
In arriving at a prime factorization, it is instructive to ask some students
to compare the result of “chipping away” by always factoring out the
smallest prime possible and “splitting asunder” by breaking the number
into two large factors. For 126, the process of chipping away corresponds
to writing
126 2 63 2 3 21 2 3 3 7,
while in splitting asunder one might write
126 9 14 3 3 2 7.
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That 258 is divisible by 2 follows from the fact that 258 200 50 8.
When 200 is divided by 2 the remainder is zero (because 200 is divisible by 2). Similarly, when 50 is divided by 2 the remainder is zero.
If 258 2 is to have a remainder, that remainder must be the result of
8 2. But because 8 is even, 8 2 has remainder 0, as does 258 2.
The above explanation as to why 258 is divisible by 2 can also be used to
explain why 258 is not divisible by 5. Both 200 and 50 are divisible by 5.
If 258 5 is to have a remainder, it must come from the units digit 8.
Since 8 1 5 3, 8 5 has a remainder of 3, as does 258 5.
In order to extend this kind of reasoning to 258 3, it is useful (for the
moment) to allow larger remainders than usual. Because 100 33 3 1,
100 3 has a remainder of 1 and 200 3 has a remainder of 2. Because
10 3 3 1, 10 3 has a remainder of 1 and 50 3 has a remainder
of 5. Finally, let us write 8 0 3 8 so that 8 3 has a “remainder” of
8. By allowing remainders of this form, we are able to conclude that the
original problem 258 3 has a remainder of 2 5 8 15, corresponding to 258 81 3 15. But now, the fact that 15 is divisible by 3 enables
us to write 258 81 3 5 3 86 3 and to conclude that 258 is
divisible by 3.
Looking back at this process, we see that:
The fact that 2 5 8 is divisible by 3 implies that 258 is
divisible by 3.
More generally, if the sum of the digits of a number is divisible by 3, so
is the number itself. Because 10n 9 has a remainder of 1 for all whole
numbers n, an almost identical argument applies to divisibility by 9. If
the sum of the digits of a number is divisible by 9, so is the number
itself. This rule is sometimes referred to as “casting out nines.”
Lesson 2.2
GCF The meaning of greatest common factor (GCF) is best conveyed by a
methodical development of factor, common factor, and then greatest common factor. That is, to find the GCF of 36 and 84, one proceeds as follows:
Factors of 36:
1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 84:
1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Common factors of 36 and 84:
1, 2, 3, 4, 6, 12
GCF of 36 and 84:
12
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Once understood, it is of interest to approach GCFs in other ways as well.
One important way of arriving at GCFs is through the prime factorizations of the numbers under consideration. For the numbers 36 and 84,
36 2 2 3 3 whereas 84 2 2 3 7.
Suppose we now denote the GCF of 36 and 84 by m and try to characterize
m in terms of its prime factorization. Since m is a divisor of both 36 and 84,
the prime factorization of m will have to appear in the product of primes for
both 36 and 84. Furthermore, since m is the greatest common factor of 36
and 84, m is the largest product of primes common to both prime factorizations. In this way we again arrive at the fact that m 2 2 3 12.
It is also possible to give GCFs a useful geometric interpretation. The fact
that 12 is a divisor of both 36 and 84 corresponds to the fact that line segments of length 12 exactly fill out line segments of length 36 and 84.
36
84
While this procedure works well for small numbers, it can be quite challenging when the numbers are large. In such situations it can be useful to
return to prime factorizations as the basis for finding LCMs.
The prime factorizations of 12 and 21 are given by
12 2 2 3 and 21 3 7.
If we denote the LCM of 12 and 21 by m, then the prime factorization of m
will have to include both of the products given above. Because the LCM of
12 and 21 is the least common multiple of these numbers, its prime factorization is the smallest product of primes that contains the two lists given
above. On this basis we again arrive at m 2 2 3 7 84. Note that
even though 3 appears in the prime factorizations of both 12 and 21, it
appears only once in the prime factorization of m.
The characterizations of GCF and LCM in terms of prime factorization
has an important consequence.
Given two whole numbers A and B, GCF(A, B) LCM(A, B) A B.
This means that a 12 12 square will exactly fill out or tile a 36 84 rectangle. The fact that 12 is the greatest common factor of 36 and 84 means that
the largest square that will tile a 36 84 rectangle has side length 12.
This geometric approach
lends itself to an important simplification. In
any tiling of a 36 84
rectangle by squares, the
side length of the tiling
squares will have to be
divisors of 36. Therefore,
they will also tile the
two 36 36 squares,
and the problem of finding the GCF becomes one of tiling the 12 36 rectangle to the right. That is, the GCF of 36 and 84 is the same as the GCF of
12 and 36. This form of reasoning is at the heart of the Euclidean algorithm
for finding GCFs.
In Lesson 2.4, an important application of GCFs arises in the simpli36
fication of fractions. In order to write in simplest form, we
36
84
84
12 3
12 7
3
7
use the GCF of 36 and 84 to write .
Lesson 2.5
LCM As with GCFs, the meaning of least common multiple (LCM) of two
numbers is best conveyed by a methodical development of multiple, common
multiple, and least common multiple. For 12 and 21, this proceeds as follows:
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144,
156, 168, 180, . . .
Multiples of 21: 21, 42, 63, 84, 105, 126, 147, 168, 189, . . .
Common multiples of 12 and 21: 84, 168, . . .
LCM of 12 and 21:
84
In the case A 12 and B 21, we have GCF(12, 21) 3 and
A B 252. We could also have found LCM(12, 21) as
12 21
GCF(12, 21)
252
3
LCM(12, 21) —— —— 84.
An important application of LCMs is in the addition and subtraction of
a
ad bc
c
fractions. While the formula always applies,
b
d
bd
there are situations in which using the LCM of b and d as a common
a
c
denominator for and can simplify the arithmetic substantially.
b
d
Lesson 2.6
COMPARING FRACTIONS As observed in the Mathematical Background
Notes for Chapter 1, our base 10 positional notation provides a convenient
way of comparing both whole numbers and decimals. However, many
12
23
students find the problem of comparing with more daunting.
17
38
To reduce the comparison of two fractions to an easier problem, one need
only write the two fractions over a common denominator. Instead of com12
23
12 38
23 17
paring with , one compares —— with —— . Now, because
17
38
17 38
38 17
12
23
12 38 456 is greater than 23 17 391, we can conclude that .
a
17
c
38
These ideas have a general formulation. In order to compare with ,
b
d
we place them both over a common denominator bd. Then the original
ad
bc
comparison is equivalent to comparing with . For positive fractions
bd
bd
this leads to the following results:
c
a
—— ≤ —— if and only if ad ≤ bc.
d
b
c
a
—— ≥ —— if and only if ad ≥ bc.
d
b
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