-- May 13 - Final Review
Question 2: For each of the following equations, find all solutions or state no solutions exists: `(a) 9x≡3mod60. (b) 34x≡5mod38. (c) 34x≡5mod57`
Solutions by Joseph, Pratik and Vishal
Solution (a): Solving `9x≡3mod60`
By Solution Existence Theorem, an equation of the form `ax ≡ b(mod n) ` has d distinct solutions modulo n, where `d = gcd(a,n)`, if `d|b`, or it has no solutions.
So, in the example `d = gcd(9, 60) = 3`.
Note, 3 divides 3. Hence, it has 3 solutions.
Solutions are given by,
` d = a * x^' + n * y^'`
This equation solves for ` x^' = 7 and y^' = -1`
`x_0` is given by `x_0 = x^'(\frac{b}{d}) mod n`
Substituting values, `x_0 = 7`
`x_0 = (7 * (\frac{3}{3})) mod 60 = 7 `
`x_i = (x_0 + i( \frac{n}{d} )) mod n,` for `i = 0,1,... `
Substituting, `i = 0, 1, 2` gives us the required 3 solutions.
Which are `7, 27, 47`
Solution (b): Solving `34x≡5mod38`
So, in the example `d = gcd(34, 38) = 2`.
But, 2 does not divide 5. Hence, it has no solutions.
Solution (c): Solving `34x≡5mod57`
So, in the example `d = gcd(34, 57) = 1`.
Note, 1 divides 5. Hence, it has exactly 1 solution.
Solutions are given by,
` d = a * x^' + n * y^'`
This equation solves for ` x^' = -5 and y^' = 3`
`x_0` is given by `x_0 = x^'(\frac{b}{d}) mod n`
Substituting values, `x_0 = -25`
`x_0 = (-5 * (\frac{5}{1})) mod 57 = 32 `
So 32 is the only solution which is also `x = -25 mod 57`
(
Edited: 2019-05-15)
----
'''Question 2:''' For each of the following equations, find all solutions or state no solutions exists: <math>(a) 9x≡3mod60. (b) 34x≡5mod38. (c) 34x≡5mod57</math>
<br/>
<br/>
Solutions by Joseph, Pratik and Vishal
<br/>
<br/>
'''Solution (a):''' Solving <math>9x≡3mod60</math>
<br/>
<br/>
By Solution Existence Theorem, an equation of the form <math>ax ≡ b(mod n) </math> has d distinct solutions modulo n, where <math>d = gcd(a,n)</math>, if <math>d|b</math>, or it has no solutions.
<br/>
<br/>
So, in the example <math>d = gcd(9, 60) = 3</math>.
<br/>
Note, 3 divides 3. Hence, it has 3 solutions.
<br/>
Solutions are given by,
<br/>
<math> d = a * x^' + n * y^'</math>
<br/>
This equation solves for <math> x^' = 7 and y^' = -1</math>
<br/>
<br/>
<math>x_0</math> is given by <math>x_0 = x^'(\frac{b}{d}) mod n</math>
<br/>
Substituting values, <math>x_0 = 7</math>
<br/>
<br/>
<math>x_0 = (7 * (\frac{3}{3})) mod 60 = 7 </math>
<br/>
<br/>
<math>x_i = (x_0 + i( \frac{n}{d} )) mod n,</math> for <math>i = 0,1,... </math>
<br/>
<br/>
Substituting, <math>i = 0, 1, 2</math> gives us the required 3 solutions.
<br/>
Which are <math>7, 27, 47</math>
----
'''Solution (b):''' Solving <math>34x≡5mod38</math>
<br/>
<br/>
So, in the example <math>d = gcd(34, 38) = 2</math>.
<br/>
But, 2 does not divide 5. Hence, it has no solutions.
----
'''Solution (c):''' Solving <math>34x≡5mod57</math>
<br/>
<br/>
So, in the example <math>d = gcd(34, 57) = 1</math>.
<br/>
Note, 1 divides 5. Hence, it has exactly 1 solution.
<br/>
Solutions are given by,
<br/>
<math> d = a * x^' + n * y^'</math>
<br/>
This equation solves for <math> x^' = -5 and y^' = 3</math>
<br/>
<br/>
<math>x_0</math> is given by <math>x_0 = x^'(\frac{b}{d}) mod n</math>
<br/>
Substituting values, <math>x_0 = -25</math>
<br/>
<br/>
<math>x_0 = (-5 * (\frac{5}{1})) mod 57 = 32 </math>
<br/>
<br/>
So 32 is the only solution which is also <math>x = -25 mod 57</math>