Unit: Number Theory
Unit Question: How can I use my knowledge of numbers to work smart, not hard?
Significant Concept: Mathematicians classify numbers and use the relationship
between numbers to solve complex problems more effectively.
Reflection Blog Posts
Answer these questions on your blog. Use examples whenever possible to
show your thinking.
1.
*The big idea for the Number Theory unit is about how mathematicians
classify numbers so that they can see relationships between numbers and
solve problems more effectively and efficiently. What are some
different ways that we classified numbers in this unit? Hint – it might
help to make a table.
Words
Explanation/Meaning
Examples
Odd Numbers
Odd numbers are numbers that 1,3,5,7,9,11,13,15,17
is not divisible by 2.
Even Numbers
Even numbers are the opposite 2,4,6,8,10,12,14,16,18
of odd numbers, So it should
be divisible by 2.
Prime Numbers
Prime numbers are numbers
2,3,5,7,11,13,17,19,23
that only have 2 factors.
,29,31
Composite Numbers
Composite numbers are
4,6,8,9,10,12,14,15,16
opposite of prime numbers so
,18,20,21,
it can have more than 2
factors but not less.
Abundant Numbers
Abundant numbers is when
12 PF: 1,2,3,4,6 = 16
you add the sum of the proper
factor for the number and the
sum is greater than the number
itself.
Deficient Numbers
Deficient is opposite of
14 PF: 1,2,7 = 10
Abundant so the sum should
be less than the number itself.
Perfect Numbers
Perfect is when the proper
6 PF: 1,2,3 = 6
factors sum is the same with
the number itself.
Square Numbers
Square numbers is when you
1x1=1
multiply 2 same numbers.
4x4=16
2x2=4
5x5=25
3x3=9
GCF (Greatest
Common Factor)
GCF is used with 2 numbers
The GCF for 24 &
and it is the biggest factor
36 is 12
that they both have the same
LCM (Least Common LCM is similar to GCF but
The LCM for 24 &
Multiple)
instead of the biggest factor
36 is 72
this is the least multiple from 2
numbers
Relatively prime is when two
Relatively Prime
numbers is
2. **What is the difference between a prime and a composite number?
Support your answer with evidence.
Prime numbers only have 2 factors and composite numbers are the opposite
of prime numbers, which means a composite number has more than two
factors.
Ex. 2, 3, 5, 7, 11, 13, 17, 19, 23 are all prime numbers because it only have 2
factors
4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21 are all composite numbers because they all
have more than one 2 factors.
3. *Why isn’t 1 a prime number?
1 only have 1 factor that means it is not a composite or a prime number.
4.
5.
**What are the factors of a number and how can you make sure that you
have found all the factors of a given number? (What strategies can you
use? What properties?)
Factors are numbers you use to multiply. For example if you want to
make a product, you have to use 2 factors to make a product.
Ex. (9 x 8(factors)) = 72
You can use GCF and LCM to find the factors and also you can double
check with the divisible by.
**What’s the difference between abundant and deficient numbers?
What are some examples of abundant numbers we use in real life?
An abundant number is when you add all the proper factors from the
number and if the sum is bigger than the number itself then it is an
abundant number if it is less than it is a deficient number.
6.
***Mark and Sarah were playing the Number Theory Riddle Game.
Mark’s secret number had 4 factors and Sarah’s secret number had 6
factors. Sarah said that her number had to be bigger than Mark’s
because it had more factors. Do you agree with her? Why or why not?
No because there are many numbers that is lower and can have a lot of
factors. For example number 14 only has 4 factors which are 1,2,7,14 and
12 has 6 factors which are 1,2,3,4,6,12.
7.
**How can you keep the definitions of factor and multiple separate in
your head? Hint: Use real life examples.
Multiple is numbers that should be higher than factors so I remember that
multiple is always going to be a bigger number and factors should be numbers
that are less bigger than the number.
8.
*What does LCM mean? How do you find it using prime factorization?
Hint – find an image or video on the web.
LCM stands for Least Common Multiple this is used when you want to find
out the smallest multiple from 2 numbers that is the same number. For
example the LCM for 18 and 24 would be 72 because that is the smallest
multiple that they both have in common. To find the LCM using prime
factorization is to first write down the prime factorization from both
numbers. Then from the two prime factorizations take which of the two
that has the most number of factors. Then just multiply all of them.
Here is the picture for this method.
Video from: http://www.youtube.com/watch?v=YZ8JdA8jVq8
9.
*What does GCF mean? How do you find it using prime factorization?
Hint – find an image or video on the web.
GCF stands for Greatest Common Factor. GCF is almost the same as LCM
but instead you have to find the biggest factor from two numbers that is
the same number. So to find the GCF using the prime factorization you have
to find the prime factorization for both numbers. Then you have to take the
numbers that have in common for example if both numbers have a 3 then
write down a 3.
Video from: http://www.youtube.com/watch?v=uE9O8N5JYB4
10. ***How can you tell whether you should use LCM or GCF to find the
answer to a story problem? Hint – use examples.
LCM: LCM is when you just multiply and get the least common multiply
GCF: GCF is when you share and you get the greatest factor.
11. **How are the divisibility rules for 3 and 9 similar Challenge: How are
the divisibility rules for 6 and 12 similar?
The divisibility rules for 3 are to add the sum of the numbers and see if it is
divisible by 3 in a smaller number. 9 is also similar to 3 because instead of 3
you add the sum and also
12. **What does it mean for numbers to be “relatively prime”? List at least 3
different pairs of numbers.
Relatively prime is when 2 numbers does not have a same factor except for
the number 1.
For example: 11 & 24 , 24 & 37, 67 & 23
True or False – decide whether each statement is true or false. Support
your answer with evidence. Remember if you say a statement is true, it has
to be true ALL the time.
13. The sum of two odd numbers is always even.
True, because if you divide and odd number into two there is always a
remainder so if you add two odd numbers then the other remainder will be
divisible to two.
14. The LCM of two different prime numbers is the product of those
numbers.
True, because prime numbers won’t have anything in common with any other
numbers except that they have 1 as their factor, so if you add with other
numbers there won’t be anything in common too.
15. The GCF of two prime numbers is always 1.
True, another word for this would be relatively prime so there is no other
way to find the factors other than 1 so the GCF would always be prime.
16. The GCF of two numbers is less than either of those numbers.
True, because factors are always going to be less than the number itself so
the GCF could not be bigger than the any of those numbers.
17. Square numbers have an odd number of factors.
True. All numbers would have a pair of factors, which are both different
numbers. But because square numbers are made out of a pair of two same
numbered factors they would have pairs of factors but one of them would
be the same number.
For example:
9 ‘s Factors: 1 x 9 and 3 x 3 Total of prime 3 because there is two 3’s just take
1.