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Factors and Multiples

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Factors and Multiples

- What are prime numbers?

A prime number is:

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

For example, 7 = 7 x 1

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

Clearly, 2 = 1 x 2

3 = 1 x 3

5 = 1 x 5

7 = 1 x 7

11 = 1 x 11

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

- What are composite numbers?

A composite number is:

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

For example, 14 = 1 x 14 and 2 x 7

So, 14 is a composite number as it has more than two factors.

EXAMPLES

- State which of the following numbers are prime:
- 46
- 19

SOLUTION:

- 46 is not a prime because 46 = 2 x 23.
- 19 is a prime since it has only two different factors, 1 and 19.

FACTORS

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

2 x 3 = 6

FactorFactor

- Explain your reasoning or give a counter example to answer
- If a number is divisible by 3, is it divisible by 9?
- If a number is divisible by 9, is it divisible by 3?

- Which of the following numbers are divisible by 4?
- Determine the remainder when the number is divided by 4?
- 14,710,816,558
- 4,328,104,292

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

- The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
- Which of these prime numbers would you have to consider as possible factors of 367 in order to determine whether 367 is a prime or composite number?
- Is 367 prime or composite?

- Which of the following numbers are prime?
- 231
- 277
- 683

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

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

- It’s odd, so it’s not divisible by 2;
- It’s not divisible by 3 (use the divisibility rule: 1 + 0 + 7 = 8, not 3 or 6 or 9)
- It’s not divisible by 5 (doesn’t end in 5 or zero)
- It’s not divisible by 7 (if it were, 107 – 7 = 100 would be divisible by 7, which we know isn’t true)
- At this point, we know it’s not prime, since we’d need to check 11 next. But 11 x 11 = 121, bigger than 107, so 11 is less than the square root of 107.

INTERACTIVE SITE FOR PRIMES AND COMPOSITES

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

Sieve of Eratosthenes

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

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

- The following example illustrates how the Sieve of Eratosthenes, can be used to find all the prime numbers that are less than 100.
- Step 1: Write the numbers 1 to 100 in ten rows.
- Step 2: Cross out 1 because 1 is not a prime.
- Step 3: Circle 2 and cross out all multiples of 2. (2, 4, 6, 8, 10, ...)
- Step 4: Circle 3 and cross out all multiples of 3. (3, 6, 9, 12, 15, ...)
- Step 5: Circle 5 and cross out all multiples of 5. (5, 10, 15, 20, ...)
- Step 6: 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 prime numbers less than 100.