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Number Theory I Junior Math Circles - University Of Waterloo Faculty Of Mathematics Worksheet With Answers Page 4

Exercise 4 Use this method to ﬁnd all the prime numbers from 1 to 100.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Write down the primes less than 100. There should be 25 of them.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Prime Factorization

One way of describing numbers is by breaking them down into a product of their

prime factors. This is called prime factorization. Every positive number can be

prime factored. By deﬁnition the prime factorization of a prime number is the num-

ber itself, and the prime factorization of 1 is 1.

Example Let’s prime factor 270 using a factor tree.

Solution First think of a factor of 270. We don’t want to use 1 or 270.

Let’s use 10.

Now what factor is paired with 10? That factor is 27, since 10

27 = 270.

270

Number Theory I Junior Math Circles - University Of Waterloo Faculty Of Mathematics Worksheet With Answers Page 4