High School Mathematics Extensions/Solutions to Problem Sets

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


 * 1) $$13 = 13 \cdot 1$$
 * 2) $$26 = 13 \cdot 2$$
 * 3) $$59 = 59 \cdot 1$$
 * 4) $$82 = 41 \cdot 2$$
 * 5) $$101 = 101 \cdot 1$$
 * 6) $$121 = 11 \cdot 11$$
 * 7) $$2187 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$$

Recursive Factorisation Exercises
Factorise using recursion.


 * 1) $$45 = 3 \cdot 3 \cdot 5$$
 * 2) $$4050 = 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5$$
 * 3) $$2187 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$$

Prime Sieve Exercises

 * 1) 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:



\begin{matrix} X & 2_p & 3_p & X & 5 &X &7& X& X& X \\ 11 & X & 13 & X& X& X&17 &X& 19& X\\ X& X& 23 & X& 25 &X&X&X&29& X\\ 31 &X& X& X& 35 &X&37& X& X& X\\ 41 & X& 43 & X& X&X&47& X& 49& X\\ \end{matrix} $$


 * 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

 * 1) $$(-1) \cdot (-5)\mod{11} = 5$$
 * 2) $$3 \cdot 7 \mod{11} = 21 = 10$$
 * 3) $$2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16 = 5$$ $$ 2^5 = 32 = 10, 2^6 = 64 = 9, 2^7 = 128 = 7$$ $$ 2^8 = 256 = 3, 2^9 = 512 = 6, 2^{10} = 1024 = 1$$ An easier list: 2, 4, 8, 5, 10, 9, 7, 3, 6, 1