# High School Mathematics Extensions/Solutions to Problem Sets

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## Contents

## Primes and Modular Arithmetic[edit]

### Factorisation Exercises[edit]

Factorise the following numbers. (note: I know you didn't have to, this is just for those who are curious)

### Recursive Factorisation Exercises[edit]

Factorise using recursion.

### Prime Sieve Exercises[edit]

- Use the above result to quickly work out the numbers that still need to be crossed out in the table below, knowing 5 is the next prime:

- The next prime number is 5. Because 5 is an unmarked prime number, and 5 * 5 = 25, cross out 25. Also, 7 is an unmarked prime number, and 5 * 7 = 35, so cross off 35. However, 5 * 11 = 55, which is too high, so mark 5 as prime ad move on to 7. The only number low enough to be marked off is 7 * 7, which equals 35. You can go no higher.

2. Find all primes below 200.

- The method will not be outlined here, as it is too long. However, all primes below 200 are:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199

### Modular Arithmetic Exercises[edit]

An easier list: 2, 4, 8, 5, 10, 9, 7, 3, 6, 1