High School Mathematics Extensions/Primes/Solutions
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HSE PrimesPrimes and Modular Arithmetic edit
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Factorisation Exercises edit
Factorise the following numbers. (note: I know you didn't have to, this is just for those who are curious)
 13 is prime
 59 is prime
 101 is prime
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 49. 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
 alternatively, 1 = 10, 5 = 6: 10 × 6 = 60 = 5× 11 + 5 = 5

An easier list: 2, 4, 8, 5, 10, 9, 7, 3, 6, 1
Notice that it is not necessary to actually
compute to find mod 11.
If you know mod 11 = 6.
You can find mod 11 = (2*( mod 11)) mod 11 = 2*6 mod 11 = 12 mod 11 = 1.
We can note that 2^{9} = 6 and 2^{10} = 1, we can calculate 6^{2} easily: 6^{2} = 2^{18} = 2^8 = 3. OR by the above method
An easier list: 6, 3, 7, 9, 10, 5, 8, 4, 2, 1.  0^{2} = 0, 1^{2} = 1, 2^{2} = 4, 3^{2} = 9,
4^{2} = 16 = 5, 5^{2} = 25 = 5, 6^{2} = 36 = 3, 7^{2} = 49 = 3,
8^{2} = 64 = 9, 9^{2} = 81 = 4, 10^{2} = 100 = 1
An easier list: 0, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1
Thus  x^{2} = 2 = 9
Just look at the list above and you'll see that
Division and Inverses Exercises edit
1.
 therefore the inverse does not exist
2.
3.
4.
0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  
1  mod 2  
1  2  mod 3  
1  3  mod 4  
1  3  2  4  mod 5  
1  5  mod 6  
1  4  5  2  3  6  mod 7  
1  3  5  7  mod 8  
1  5  7  2  4  8  mod 9  
1  7  3  9  mod 10  
1  6  4  3  9  2  8  7  5  10  mod 11  
1  5  7  11  mod 12  
1  7  9  10  8  11  2  5  3  4  6  12  mod 13  
1  5  3  11  9  13  mod 14  
1  8  4  13  2  11  7  14  mod 15  
1  11  13  7  9  3  5  15  mod 16  
1  9  6  13  7  3  5  15  2  12  14  10  4  11  8  16  mod 17  
1  11  13  5  7  17  mod 18  
1  10  13  5  4  16  11  12  17  2  7  8  3  15  14  6  9  18  mod 19 
Coprime and greatest common divisor Exercises edit
1.
 1.
smaller  larger 

5050  5051 
1  5050 
0  1 
 5050 and 5051 are coprime
 2.
smaller  larger 

59  78 
19  59 
2  19 
1  2 
0  1 
 59 and 79 are coprime
 3.
smaller  larger 

111  369 
36  111 
3  36 
0  3 
 111 and 369 are not coprime
 4.
smaller  larger 

2021  4032 
2011  2021 
10  2011 
1  10 
0  1 
 2021 and 4032 are coprime
2. We first calculate the gcd for all combinations
smaller  larger 

15  510 
0  15 
smaller  larger 

15  375 
0  15 
smaller  larger 

375  510 
135  375 
105  135 
30  105 
15  30 
0  15 
 The gcd for any combination of the numbers is 15 so the gcd is 15 for the three numbers.
Diophantine equation Exercises edit
1.
 There is no solution, because can never become an integer.
2.
 We choose d=1, then x=26.
3.
 (a)
smaller  larger  PQ 

33  101  3 
2  33  16 
1  2  2 
0  1 
3  16  2  

0  1  3  49  101  1  0  1  16  33 
 (b) To be added
4.
 (a)
smaller  larger  PQ 

17  317  18 
11  17  1 
6  11  1 
5  6  1 
1  5  5 
0  1 
18  1  1  1  5  

0  1  18  19  37  56  317  1  0  1  1  2  3  17 
 (b) To be added
Chinese remainder theorem exercises edit
1.
Question 1 edit
Show that the divisibleby3 theorem works for any 3 digits numbers (Hint: Express a 3 digit number as 100a + 10b + c, where a, b and c are ≥ 0 and < 10)
Solution 1 Any 3 digits integer x can be expressed as follows
 x = 100a + 10b + c
where a, b and c are positive integer between 0 and 9 inclusive. Now
if and only if a + b + c = 3k for some k. But a, b and c are the digits of x.
Question 2 edit
"A number is divisible by 9 if and only if the sum of its digits is divisible by 9." True or false? Determine whether 89, 558, 51858, and 41857 are divisible by 9. Check your answers.
Solution 2 The statement is true and can be proven as in question 1.
Question 4 edit
The prime sieve has been applied to the table of numbers above. Notice that every number situated directly below 2 and 5 are crossed out. Construct a rectangular grid of numbers running from 1 to 60 so that after the prime sieve has been performed on it, all numbers situated directly below 3 and 5 are crossed out. What is the width of the grid?
Solution 4 The width of the grid should be 15 or a multiple of it.
Question 6 edit
Show that n  1 has itself as an inverse modulo n.
Solution 6
 (n  1)^{2} = n^{2}  2n + 1 = 1 (mod n)
Alternatively
 (n  1)^{2} = (1)^{2} = 1 (mod n)
Question 7 edit
Show that 10 does not have an inverse modulo 15.
Solution 7 Suppose 10 does have an inverse x mod 15,
 10x = 1 (mod 15)
 2×5x = 1 (mod 15)
 5x = 8 (mod 15)
 5x = 8 + 15k
for some integer k
 x = 1.6 + 3k
but now x is not an integer, therefore 10 does not have an inverse