Chapter 1 | Review of Basic Arithmetic
20. Estimate the following by first rounding the numbers to the nearest ten:
a. 63 # 59
b. 35 # 97
c. 246 # 45
d. 78 ÷ 19
e. 245 ÷ 45
f. 38 ÷ 8
21. A prepaid phone card to Singapore costs $5 for 3 hours. Find the cost per minute rounded to the nearest cent.
22. A prepaid phone card to China costs $2.50 for 1 hour. Find the cost per minute rounded to the nearest cent.
23. 55.25% of the employees of a large software company are engineers, 35.40% are project managers and team leads,
2.40% are senior managers, and the rest are administrative staff. Express these percentages rounded to the nearest
percentage without decimals.
24. Amy invested 4.20% of her savings in bonds, 32.65% in stocks, 25.55% in mutual funds, and the balance in her
brother's business. Express these percentages rounded to the nearest percentage without decimals.
1.2 |
Factors and Multiples
Factors of a number are whole numbers that can divide the number with no remainder.
■■For example, factors of 12 are 1, 2, 3, 4, 6, and 12. We can express factors of a
number by showing how the product of two factors results in the number.
12 = 1 # 12
12 = 2 # 6
12 = 2 # 2 # 3
12 = 3 # 4
Multiples of a number are whole numbers that can be divided by the number with no
remainder. Multiples of a number can be expressed as the product of the number and a whole
number.
■■For example, multiples of 10:
10, (10+10), (10+10+10), (10+10+10+10), (10+10+10+10+10) ...
10,
20,
10, (10#2),
30,
40, 50
(10#3),
(10#4), (10#5)
Therefore, multiples of 10 are 10, 20, 30, 40, 50 …
­­­ rime Numbers and Composite Numbers
P
A prime number is a whole number that has only two factors: 1 and the number itself;
i.e., prime numbers can be divided only by 1 and the number itself.
■■For example, 7 is a prime number because it only has two factors: 1 and 7.
Example 1.2(a)
Identifying Prime Numbers
Identify all the prime numbers less than 25.
Solution
The prime numbers less than 25 are 2, 3, 5, 7, 11, 13, 17, 19, and 23.
A composite number is a whole number that has at least one factor other than 1 and the number
itself; i.e., all whole numbers that are not prime numbers are composite numbers.
0 and 1 are neither
prime numbers nor
composite numbers.
■■For example, 8 is a composite number because it has
more than 2 factors: 1, 2, 4, and 8.
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Chapter 1 | Review of Basic Arithmetic
Example 1.2(b)
Identifying Composite Numbers
Identify all the composite numbers less than 25.
Solution
All the composite numbers less than 25 are
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, and 24.
Example 1.2(c)
Finding Factors of Prime Numbers
Find all the factors of 13.
Solution
1 and 13 are the only factors of 13.
Example 1.2(d)
Finding all Factors of Composite Numbers
Find all the factors of:
(i) 12
(ii) 20
(i) The factors of 12 are 1, 2, 3, 4, 6, and 12.
(ii) The factors of 20 are 1, 2, 4, 5, 10, and 20.
Solution
Example 1.2(e)
Finding the Prime Factors of Composite Numbers
Find all the prime factors of 24.
Solution
All the factors of 24 are: 2, 3, 4, 6, 8, 12, and 24.
In the above factors, only 2 and 3 are prime numbers.
Therefore, the prime factors of 24 are 2 and 3.
Least or Lowest Common Multiple (LCM)
­­­
The Lowest Common Multiple (LCM) of two or more whole numbers is the smallest multiple
that is common to those numbers.The LCM can be determined from one of the following methods:
Method 1
(1) First, select the largest number and check to see if it is divisible by all the other numbers.
If it divides, then the largest number is the LCM.
(2) If the largest number does not divide the other numbers then find a multiple of the largest
number that is divisible by all the other numbers.
(3) If none of the multiples of the largest number are divisible by all the other numbers (i.e.,
if the numbers have no common factors), then the LCM of the numbers is the product of
all the numbers.
Method 2
(1) Find the prime factors of each of the numbers and list the different prime numbers (using
a factor tree as shown in the example that follows).
(2) Count the number of times each different prime number appears in each of the
factorizations.
(3) Find the largest of these counts for each prime number.
(4) List that prime number as many times as you counted for it in step 3.
The LCM is the product of all the prime numbers listed.
Chapter 1 | Review of Basic Arithmetic
Example 1.2(f)
Finding the Lowest Common Multiple
Find the LCM of the following:
(i) 3, 6, and 18
(ii) 9 and 15
(iii) 3, 5, and 8
Solution
(i) 3, 6, and 18 Method 1
(1) The largest number, 18, is divisible by both 6 and 3.
Therefore, 18 is the LCM of 3, 6, and 18.
Method 2
(ii) 9 and 15
Method 1
(1) The largest number, 15, is not divisible by 9.
(2) Multiples of 15 are 15, 30, 45…
45 is divisble by 9.
Therefore, 45 is the LCM of 9 and 15.
Method 2
■■ If the last digit of the
number is 0, 2, 4, 6, or
8, then the number is
divisible by 2.
■■ If the sum of digits of
the numbers is divisible
by 3, then the number
is divisible by 3.
■■ If the last digit of the
number is 0 or 5, then
the number is divisible
by 5.
(iii) 3, 5, and 8
Method 1
(1) The largest number, 8, is not divisible by 3 and 5.
(2) Multiples of 8 are 16, 24, 32, 40, 48, 56, 64…
None of the multiples listed above are divisible by 3 and 5.
(3) Since 3, 5, and 8 have no common factors, the LCM is the product
of all the numbers: 3 # 5 # 8 = 120.
Therefore, 120 is the LCM of 3, 5, and 8.
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Chapter 1 | Review of Basic Arithmetic
Solution
continued
Method 2
Least or Lowest Common Denominator (LCD)
The Lowest Common Denominator (LCD) of a set of two or more fractions is the smallest
whole number that is divisible by each of the denominators. It is the least common multiple (LCM)
of the denominators of the fractions.
In performing addition and subtraction of fractions, it is necessary to find the equivalent fraction
using the least common denominator. The best choice for a common denominator is the LCD,
because it makes any further simplification easier.
Example 1.2(g)
Finding the Lowest Common Denominator
7
4
Find the LCD of and 15 .
9
Solution
4
7
and
is the same as the LCM of the denominators 9 and 15. It is
9
15
4
7
the same as in Example 1.2(f)(ii). Therefore, the LCD of and
is 45.
9
15
­­­
The LCD of the fractions
Highest Common Factor (HCF)
The factors that are common for two or more numbers are called common factors of those
numbers.
The Highest Common Factor (HCF) of two or more numbers is the largest common number that
divides the numbers with no remainder. In other words, the HCF is the largest of all the common
factors. HCF is useful when simplifying fractions. For this purpose, HCF is also known as the
Greatest Common Divisor (GCD).
Example 1.2(h)
Finding the Common Factors
Find the common factors of 12 and 18.
Solution
The factors of 12 are 1, 2, 3, 4, 6, 12.
The factors of 18 are 1, 2, 3, 6, 9, 18.
The common factors are: 2, 3, and 6.
1 is a factor that is common to all
numbers but is not included in the
list of common factors.