Prime and Composite Numbers
Factors
What is a prime number?
Is this number prime?
Teaching Prime Numbers
• Elementary school children can learn prime
numbers, and the process can become an
exciting challenge.
• As teachers, we can do much to make this a
good learning experience for our students.
• We begin simply, by reviewing multiplication
Multiplication
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2 x 3 = ____
Yes, we start with an easy one.
We know that 2 x 3 = 6
We remind our students that:
2 and 3 are factors
6 is the product
Showing Factors
• Here are 6 blocks
• We can arrange them to make a rectangle
That is 3 blocks long and 2 blocks wide
This shows that 2 and 3 are factors.
Finding the factors of 6
• We have already shown that 2 and 3 are
factors of 6.
• Are there any others?
• We know that 1 x 6 = 6
• So, 1 and 6 are also factors of 6.
• That means 1, 2, 3, and 6 are factors.
• Are there any more?
Other Factors of 6
• Here are 6 blocks
• We can arrange them to make a rectangle
That is 6 blocks long and 1 block wide
This shows that 1 and 6 are also factors of 6.
• That means 1, 2, 3, and 6 are factors.
• Are there any more?
• The only other numbers we can try are 4 and
5.
• Let’s see if we can make rectangles.
Testing other factors
• Can we make a rectangle with 6 blocks
that is 4 blocks long?
• No, we can’t. We would need two more
blocks to make a rectangle.
The Factors of 6
We can make a rectangle that is 2 x 3 blocks.
That means 2 and 3 are factors of 6.
We can make a rectangle that is 1 x 6 blocks.
That means 1 and 6 are factors of 6.
Those are the only rectangles with 6 blocks.
We know that 1, 2, 3, and 6
are the only factors of 6.
Making a List
• The teacher helps students make a list of
every factor for every number up to 10.
• The factors of 6 have already been
determined.
Beginning the Factor List
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1
2
3
4
5
6 1, 2, 3, 6
7
8
9
10
A Factor List
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1
2
3
4
5
6
7
8
9
10
1
1, 2
1, 3
1, 2, 4
1, 5
1, 2, 3, 6
1, 7
1, 2, 4, 8
1, 3, 9
1, 2, 5, 10
What Do We Notice?
• The only number with only one factor is “1.”
Every number greater than 1 has at least two
factors, 1 and the number itself: these are the
identity factors, 1 x n = n.
• If there are two factors, and only two factors,
then the number is prime. If there are more
than two factors, the number is composite.
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Finding the Primes
• In looking at the factor list, we see that 2, 3, 5,
and 7 are prime numbers.
• For composite numbers (4, 6, 8, 9, and 10) the
smallest factor (other than 1) is always a
prime number.
• It is called the Smallest Prime Factor (SPF).
Prime and Composite Numbers
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1
2
3
4
5
6
7
8
9
10
Neither
Prime
Prime
Composite (2)
Prime
Composite (2)
Prime
Composite (2)
Composite (3)
Composite (2)
The Next Steps
• We could continue this process of finding all
the factors of a number, and noticing whether
or not there are exactly two factors.
• This is a good thing for children to do – up to
20 or 25.
However
• There is a much easier method ...
The Sieve of Eratosthenes
• Eratosthenes lived over 2000 years ago.
• He invented a method for finding prime
numbers that is still used today.
• The method is brilliant – and very simple.
• Start with a 100-chart.
• Cross out 1.
Blot out the number 1
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Look at the next number, 2.
It has no color.
So, we circle 2.
Then we cross out every multiple of 2.
Circle 2 and color its multiples
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Look at the next number, 3.
It has no color.
So, we circle 3.
Then we cross out every multiple of 3.
Circle 3 and color its multiples
• Look at the next number, 4.
• It DOES HAVE color.
So, we ignore it.
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Look at the next number, 5.
It has no color.
So, we circle 5.
Then we cross out every multiple of 5.
Circle 5 and color its multiples
• Look at the next number, 6.
• It DOES HAVE color.
So, we ignore it.
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Look at the next number, 7.
It has no color.
So, we circle 7.
Then we cross out every multiple of 7.
Circle 7 and color its multiples
The Sieve is Complete
• We notice that 8, 9, and 10 have already been
crossed out.
• We circle 11 and begin to cross out its
multiples.
• We notice that every multiple of 11 has
already been crossed out!
• We have found all the prime numbers!
Let’s Show it in Color
• It looks better in color.
• We can also see more mathematical ideas.
A Colorful Sieve
What Tests Do We Need?
• So we do need to test 2, 3, 5, and 7.
• Do we need to test any other numbers?
• In particular, do we need to test whether a number is
divisible by 11?
• 11, is a prime number.
• The other multiples are: 22, 33, 44, 55, 66, 77, 88, and
99.
• 22, 44, 66, and 88 are also divisible by 2;
• 33 and 99 are divisible by 3;
• 55 is divisible by 5;
• 77 is divisible by 7.
The Multiples of 11
• 11, is a prime number.
• The other multiples are:
22, 33, 44, 55, 66, 77, 88, and 99.
• 22, 44, 66, and 88 are also divisible by 2
• 33 and 99 are divisible by 3
• 55 is divisible by 5
• 77 is divisible by 7
• Divisibility Tests
• After a thorough discussion of the Sieve of
Eratosthenes, students will know that 2, 3, 5, and
7 are prime numbers, and that 4, 6, 8, 9, and 10
are composite numbers. Furthermore, only four
divisibility tests need to be performed for any
number in the 100-chart: 2, 3, 5, and 7. If a
number is divisible by any of those four factors,
the number is composite. If it is not divisible,
then it must be prime.
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The Easy Divisibility Tests
• A number divisible by:
2 if it is even
5 if if ends in 0 or 5
The Divisibility Test for 3
• A number is divisible by 3:
if the sum of the digits is 3, 6, or 9.
For 87, we add the digits: 8 + 7 = 15.
Then we continue the process: 1 + 5 = 6,
and we see that 87 is divisible by 3.
For 71 , we add the digits: 7 + 1 = 8.
and we see that 71 is NOT divisible by 3.
The Multiples of 7
• There is no easy divisibility test for 7.
• It is useful to recall the multiples of 7:
14 21 28 35 42 49 56 63 70
77 84 91 98
The Divisibility Test for 7
But we don’t have to remember all.
After we test for 2, 3, and 5;
Only three multiples of 7 are left:
49 77 91
Divisibility Tests: Summary
2
3
5
7
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Even
Digit sum is 3, 6, or 9
Ends in 0 or 5
49, 77, 91
All other multiples of 7 have been eliminated
Example
82 is even
This means that is has a factor of 2.
This means there are more than two factors.
[We know that 1 and 82 are factors.]
Therefore, 82 is a composite number.
Example
57 has a digit sum:
5 + 7 = 12
1+2=3
This means that is has a factor of 3.
This means there are more than two factors.
Therefore, 57 is a composite number.
Example
Test 91
Is odd, so 2 is not a factor
Digit sum = 10, so 3 is not a factor
Ends in 1, so 5 is not a factor,
But 91 is divisible by 7 [7 x 13 = 91]
Therefore, 91 is a composite number.
Example
Test 71
2
3
5
7
Odd
Digit sum = 7 + 1
Ends in 1
Not divisible by 7
71 is not divisible by 2, 3, 5 or 7;
it is prime
Summary
The prime numbers less than 10 are 2, 3, 5, or 7
For any number from 10 to 100
If it is divisible by 2, 3, 5, or 7
then it is composite.
Otherwise the number is prime.