Download Math 101 Study Session Quiz 1 Chapter 3 Sections 1 through 4

Math 101 Study Session Quiz 1 Chapter 3 Sections 1
through 4
July 28, 2016
Chapter 3 Section 1: The Least Common Multiple and
Greatest Common Factor
Natural number factors of a number divide that number evenly (there is no remainder).
Note:
• 1 is a factor of every number
• 2 is a factor of a number if the the last digit of the number is either 2, 4, 6, 8, or 0.
Ex: 436 ends in 6, therefore 2 is a factor of 436.
• 3 is a factor of a number if the sum of the digits is divisible by 3.
Ex: The sum of the digits of 489 is 4 + 8 + 9 = 21. 21 is divisible by 3, therefore 3 is
a factor of 489.
• 4 is a factor of a number if the last two digits of the number is divisible by 4.
Ex: 556 ends in 56. 56 is divisible by 4, therefore 4 is a factor of 556.
• 5 is a factor of a number if the the last digit of the number is either 5 or 0.
Ex: 520 ends in a 0 therefore 5 is a factor of 520.
• 6 is a factor of a number is 2 and 3 are factors of the number.
Ex: 522 ends in a 2 so 2 is a factor of 522 and 5 + 2 + 2 = 9 is divisible by 3 so 3 is a
factor of 522 and since 2 and 3 are factors of 522, 6 is also a factor of 522.
• 8 is a factor of a number if the last 3 digits of the number is divisible by 8.
Ex: The last three digits of 77184 is 184 which is divisible by 8. Therefore 8 is a factor
of 77184.
• 9 is a factor of a number if the sum of the digits of the number is divisible by 9.
Ex: The sum of the digits of 6993 is 6 + 9 + 9 + 3 = 27. 27 is divisible by 9, therefore
9 is a factor of 6993.
1
2
• 10 is a factor of a number if the number ends in 0. (Please note: 2 and 5 are also
factors of a number that ends in 0).
Ex: 13860 ends in a 0 so 10 is a factor of 13860.
• A number is a factor of itself.
3
A prime number is a natural number greater than 1 that has exactly two natural number
factors, 1 and the number itself.
A number that is not prime is a composite number.
The prime factorization of a number is the expression of the number as a product of its
prime factors.
The least common multiple (LCM) is the smalles common multiple of two or more numbers.
The greatest common multiple (GCF) is the largest common factor of two or more
numbers.
Chapter 3 Section 2: Introduction to Fractions
A fraction can represent the number of equal parts of a whole.
A proper fraction is a fraction whose numerator is smaller than its denominator. The
decimal value of a proper fraction is a number less that one
A mixed number is a number greater than 1 with a whole-number part and a fractional
part.
An improper fraction is a fraction whose numerator is larger than or equal to its denominator. The decimal value of a improper fraction is a number greater than or equal to 1. The
numerator of an improper fraction is greater than or equal to the denominator.
Writing an improper fraction as a mixed number of
whole number.
4
1
Writing a mixed number as an improper fraction.
Chapter 3 Section 3 Writing Equivalent Fractions
Writing the simplest form of a fraction means writing it so that the numerator and denominator have no common factors other than 1.
Equal fractions with different denominators are called equivalent fractions
To find the order relation between two fractions with the same denominator, compare
the numerators. The fraction that has the smaller numerator is the smaller fraction.
Chapter 3 Section 4 Multiplication and Division of Fractions
To multiply two (or more) fractions, multiply the numerators together and multiply the
denominators together:
a·c
a c
· =
b d
b·d
To divide two fractions, we use “Keep Change Flip”
a c
a d
÷ = ·
b d
b c
5
Below are some examples for us to try with solutions at the end of the study
guide:
Solve each of the following:
1. Find the prime factorization of 40.
2. Find all the factors of 44.
3. Find the least common multiple (LCM) of 4, 18.
4. Find the least common multiple (LCM) of 20, 32.
5. Find the greatest common factor (GCF) of 15, 100.
6. Find the greatest common factor (GCF) of 12, 18.
5
as an improper fraction.
16
15
8. Write
as a mixed or whole number.
4
1 3
9. Multiply 2 · 9
2 5
15 8
10. Multiply
·
32 9
7. Write 5
Solutions to the Study Guide Questions
Solution to 1:
Note:
40 = 4 · 10
=2·2·2·5
= 23 · 5
Hence, the prime factorization of 40 is 23 · 5.
Solution to 2:
Note that 1 and 44 are factors of 44. Also note that 44 ends in a 4 so 2 and 4 are factors
of 44. Finally, note that 44 · 1 = 44, 2 · 22 = 44, 4 · 11 = 44. Hence, the factors of 44 are
1, 2, 4, 11, 22, 44.
Solution to 3:
6
2
3
4= 2·2
18 =
2
3·3
To find the LCM of 4 and 18 multiply the largest product of each prime number. Hence, the
LCM of 4 and 18 is 2 · 2 · 3 · 3 = 36.
Solution to 4:
2
5
20 =
2·2
5
32 = 2 · 2 · 2 · 2 · 2
To find the LCM of 20 and 32 multiply the largest product of each prime number. Hence,
the LCM of 20 and 32 is 2 · 2 · 2 · 2 · 2 · 5 = 160.
Solution to 5:
2
3
5
15 =
3
5
100 = 2 · 2
5·5
To find the GCF we multiply the smallest product of each prime number in each column
that does not have a blank. Notice that there are blanks in the 2 column and the 3 column.
Hence, the GCF of 15 and 100 is 5.
Solution to 6:
2
3
12 = 2 · 2
3
18 =
2
3·3
To find the GCF we multiply the smallest product of each prime number in each column
that does not have a blank. Hence the GCF of 12 and 18 is 2 · 3 = 6.
Solution to 7:
5
5
5 · 16 + 5
=
16
16
80 + 5
=
16
85
=
16
7
Solution to 8:
15
To write
as a mixed or whole number, we use long division:
4
3
4 15
12
3
Hence
15
3
=3 .
4
4
Solution to 9:
2·2+1 9·5+3
1 3
·
2 ·9 =
2 5
2
5
5 48
= ·
2 5
5 · 48
=
2·5
2·2·2·2·3·5
=
2·5
= 24
Solution to 10:
15 · 8
15 8
· =
32 9
32 · 9
2·2·2·3·5
=
2·2·2·2·2·3·3
5
=
12