Factors and Factors and MultiplesMultiples

What are prime numbers?What are prime numbers?

If a number has only two If a number has only two different factors, 1 and itself, different factors, 1 and itself, then the number is said to then the number is said to be prime.be prime.

For example, 7 = 7 x 1For example, 7 = 7 x 1

7 is a prime number since it has only two 7 is a prime number since it has only two different factors.different factors.

Clearly, 2 = 1 x 2Clearly, 2 = 1 x 2 3 = 1 x 33 = 1 x 3 5 = 1 x 55 = 1 x 5 7 = 1 x 77 = 1 x 7 11 = 1 x 1111 = 1 x 11

Therefore 2, 3, 5, 7, 11… are all prime Therefore 2, 3, 5, 7, 11… are all prime numbers.numbers.

What are composite numbers?What are composite numbers?

A number that has more A number that has more than two factors is called than two factors is called a composite number.a composite number.

For example, 14 = 1 x 14 and 2 x 7For example, 14 = 1 x 14 and 2 x 7So, 14 is a composite number as it So, 14 is a composite number as it

has more than two factors.has more than two factors.

State which of the following State which of the following numbers are prime:numbers are prime:

a)a) 4646b)b) 1919

46 is not a prime because 46 46 is not a prime because 46 = 2 x 23.= 2 x 23.

19 is a prime since it has only 19 is a prime since it has only two different factors, 1 and two different factors, 1 and 19.19.

‘‘Factors’ are the numbers Factors’ are the numbers you multiply to get you multiply to get another number:another number:

2 x 3 = 62 x 3 = 6

FactorFactor FactorFactor

Explain your reasoning or give a counter Explain your reasoning or give a counter example to answerexample to answer

• If a number is divisible by 3, is it divisible by 9?If a number is divisible by 3, is it divisible by 9?

• If a number is divisible by 9, is it divisible by 3?If a number is divisible by 9, is it divisible by 3?

Which of the following numbers are Which of the following numbers are divisible by 4?divisible by 4?

Determine the remainder when the number is Determine the remainder when the number is divided by 4?divided by 4?

• 14,710,816,55814,710,816,558

• 4,328,104,2924,328,104,292

The first 10 prime numbers are 2, 3, 5, 7, The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.11, 13, 17, 19, 23 and 29.

• Which of these prime numbers would you have to Which of these prime numbers would you have to consider as possible factors of 367 in order to consider as possible factors of 367 in order to determine whether 367 is a prime or composite determine whether 367 is a prime or composite number?number?

• Is 367 prime or composite?Is 367 prime or composite?

Which of the following numbers are prime?Which of the following numbers are prime?

• 231231

• 277277

• 683683

To test if a number is prime or To test if a number is prime or composite by hand, the easiest composite by hand, the easiest thing to do is test if its divisible thing to do is test if its divisible by prime numbers. If none of by prime numbers. If none of them divide it, once the numbers them divide it, once the numbers you’re dividing by get bigger than you’re dividing by get bigger than the square root of the number the square root of the number you’re testing, you’re done and you’re testing, you’re done and know it’s prime.know it’s prime.

For example, here’s how you would test For example, here’s how you would test if 107 is prime:if 107 is prime:

It’s odd, so it’s not divisible by 2;It’s odd, so it’s not divisible by 2; It’s not divisible by 3 It’s not divisible by 3 (use the divisibility rule: 1 + 0 (use the divisibility rule: 1 + 0

+ 7 = 8, not 3 or 6 or 9)+ 7 = 8, not 3 or 6 or 9) It’s not divisible by 5 It’s not divisible by 5 (doesn’t end in 5 or zero)(doesn’t end in 5 or zero)

It’s not divisible by 7 It’s not divisible by 7 (if it were, 107 – 7 = 100 (if it were, 107 – 7 = 100 would be divisible by 7, which we know isn’t true)would be divisible by 7, which we know isn’t true)

At this point, we know it’s not prime, since At this point, we know it’s not prime, since we’d need to check 11 next. But 11 x 11 = we’d need to check 11 next. But 11 x 11 = 121, bigger than 107, so 11 is less than the 121, bigger than 107, so 11 is less than the square root of 107.square root of 107.

INTERACTIVE SITE FOR INTERACTIVE SITE FOR PRIMES AND PRIMES AND COMPOSITESCOMPOSITES

http://www.321know.com/fra63ax2.htmhttp://www.321know.com/fra63ax2.htm

Sieve of EratosthenesSieve of Eratosthenes

A A prime numberprime number is a whole number that has exactly two factors, 1 and is a whole number that has exactly two factors, 1 and itself. itself.

We can use the We can use the Sieve of Eratosthenes Sieve of Eratosthenes to find out whether a number is to find out whether a number is prime or composite.prime or composite.

   The following example illustrates how the Sieve of Eratosthenes, can The following example illustrates how the Sieve of Eratosthenes, can

be used to find all the prime numbers that are less than 100. be used to find all the prime numbers that are less than 100. Step 1Step 1: Write the numbers 1 to 100 in ten rows. : Write the numbers 1 to 100 in ten rows. Step 2:Step 2: Cross out 1 because 1 is not a prime. Cross out 1 because 1 is not a prime. Step 3:Step 3: Circle 2 and cross out all multiples of 2. (2, 4, 6, 8, 10, ...) Circle 2 and cross out all multiples of 2. (2, 4, 6, 8, 10, ...) Step 4:Step 4: Circle 3 and cross out all multiples of 3. (3, 6, 9, 12, 15, ...) Circle 3 and cross out all multiples of 3. (3, 6, 9, 12, 15, ...) Step 5Step 5: Circle 5 and cross out all multiples of 5. (5, 10, 15, 20, ...) : Circle 5 and cross out all multiples of 5. (5, 10, 15, 20, ...) Step 6Step 6: Circle 7 and cross out all multiples of 7. (7, 14, 21, 28, ...) : Circle 7 and cross out all multiples of 7. (7, 14, 21, 28, ...) Circle all the numbers that are not crossed out and they are the Circle all the numbers that are not crossed out and they are the

prime numbers less than 100. prime numbers less than 100.