DEPARTMENT OF MATHEMATICS
DATE:
FUNDAMENTALS OF NUMBER THEORY
SEPTEMBER 2ND 2022
GRADE 6TH
SCHOOL YEAR 2022-2023
TEACHER: RODRIGO MOSQUERA G.
EXPLORATION
On Thursday morning, the local radio station held a call-in contest.
• Every third caller won a T-shirt.
• Every seventh caller won a baseball cap.
In 50 calls, which callers won a T-shirt? A baseball cap? Both prizes?
WHAT IS NUMBER THEORY?
Number theory, branch of Mathematics concerned with properties of the
positive integers (1, 2, 3, …). Sometimes called “higher arithmetic,” it is
among the oldest and most natural of mathematical pursuits.
[Taken from Britannica, Science, Number Theory]
SOME BASIC DEFINITIONS
A multiple of a number is a number that can be divided
evenly by another number without a remainder (or, the
multiples of a number are the numbers that we get after
multiplying the number by a whole number).
Example: The multiples of 6 are:
𝑀6 = {6, 12, 18, 24, 30, 36, … }
[Taken from Math Makes Sense 6, Ed. Pearson]
A factor is one of two or more numbers that divides a given number without a remainder (that is, is divides the given number exactly).
Example: The factors of 15 are 𝐷15 = {1, 3, 5, 15}
Divisibility rules. A divisibility test is an easy way to identify
whether the given number is divided by a fixed divisor without
actually performing the division process. If a number is completely
divided by another number, then the quotient should be a whole
number and the remainder should be zero.
A prime number is a number which has exactly two factors: 1 and the number itself.
A composite number has more than two factors, which means apart from getting divided by 1 and the number itself, it can also be divided by at least
one positive integer. 1 is not a prime neither a composite number.
PRACTICE
1.
List the first 5 multiples of each number.
a.
4
d.
6
b.
9
e.
12
c.
25
f.
100
2.
Which numbers below are multiples of 6? What strategy did you use to find out?
3.
Use the Venn diagram on Fig. 3 to sort these numbers: 45, 24, 52, 30, 66, 15, 85, 90, 72, 60, 20, 38.
What can you say about the numbers in the overlap?
4.
Use the hundred chart (Fig. 4). Color the multiples of 7. Circle the multiples of 3. What are the common multiples of 7 and 3 on the chart?
Fig. 3
Fig. 4
5.
6.
Find the first 3 common multiples of each set of numbers.
a.
2, 3, and 9
c.
4, 5, and 10
b.
2, 3, and 5
d.
6, 7, and 8
Bethany wears jeans every 2 days. She wears running shoes every 3 days. If she wears jeans with running shoes on May 1, what are the next 3
dates on which she will wear both jeans and running shoes?
7.
List all the factors of each number. Then, classify each of them as prime or composite.
a.
15
c.
27
e.
8
b.
18
d.
34
f.
5
8.
Write 2 numbers less than 50 that have exactly 3 factors. Which of these numbers are composite numbers? Explain your answer.
9.
At Green Middle School, there are 945 students. Is it possible to split up all of the students into groups of three? Into groups of six?
10. Eggs are packaged in cartons of 12. Which of these numbers of eggs can be packaged in full cartons? How do
you know?
a.
96
c.
60
b.
56
d.
74
11. Between 20 and 28 students signed up for the chess club. The students could not be divided exactly into
groups of 2, 3, 4, or 5. How many students signed up for the chess club? Show your work.
12. Use the diagram below to sort the numbers from 2 to 30.
13. Arianna and Esteban play a game with this spinner. Arianna gets a point if the pointer lands on a prime number.
Esteban gets a point if the pointer lands on a composite number. The first person to get 20 points wins. Who is
more likely to win? How do you know?