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Number Theory I Junior Math Circles - University Of Waterloo Faculty Of Mathematics Worksheet With Answers Page 3

Prime Numbers

Deﬁnitions:

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

A composite number it is a number with factors other than 1 and itself.

Exceptional Case: The number 1 is neither prime nor composite by deﬁnition.

Exercise 3 Determine whether the following numbers are prime or composite. If it

is composite, list out its factors.

1) 19

2) 22

prime

composite - 1, 2, 11, 22

3) 37

4) 51

prime

composite - 1, 3, 17, 51

Are there any even numbers that are prime?

Yes, 2 is the only prime number that is even. Think about why.

How can we ﬁnd prime numbers?

We can ﬁnd them using the method called Sieve of Eratosthenes.

Instructions:

1. Cross oﬀ 1 since it is not prime.

2. Find the ﬁrst prime number (2) and circle it.

3. Go through the rest of the chart and cross oﬀ all of the multiples of two.

4. Go back to the beginning of the chart and ﬁnd the ﬁrst number that is not

crossed oﬀ. Circle it, it is prime. Then go through the rest of the chart,

crossing oﬀ the multiples of this number.

5. Repeat step 4, until every number in the chart is either circled or crossed oﬀ.

6. The numbers that are circled are the prime numbers.

Number Theory I Junior Math Circles - University Of Waterloo Faculty Of Mathematics Worksheet With Answers Page 3