Solve for : y = 14x + 4 (mod 51) Step 1. subtract 4 y- 4 154 Step 2. Find the multiplicative inverse i of 14 We need to solve for in the following congruence: 14. i 1✔ (mod 51) Consider: 1 = 52 = 103 = 154 = 205 = 256 (mod 51) ✔ = 14x + 4- 4 Now we can solve for ¿ in the regular equation: 14 i 154 154 154 ✔from both the left and right sides of the equation ✓ to get that the inverse of 14 is -1 x= 154 ×(y- x(y- 4 Step 3. Multiply both the left and right side of the equation by 154 ×(y- x(y- 4 So the solved form for x is: (mod 51) x (y- 4 ♦ ✔ is the first number that is a multiple of 14 and is equivalent to 1 in mod 51. ✔in mod 51. X in mod 51. = 154 = x (mod 51) ) (mod 51) Solving a Linear Mod Congruence X x 14x (mod 51) by using the property of multiplicative inverses in mod 51
Solve for : y = 14x + 4 (mod 51) Step 1. subtract 4 y- 4 154 Step 2. Find the multiplicative inverse i of 14 We need to solve for in the following congruence: 14. i 1✔ (mod 51) Consider: 1 = 52 = 103 = 154 = 205 = 256 (mod 51) ✔ = 14x + 4- 4 Now we can solve for ¿ in the regular equation: 14 i 154 154 154 ✔from both the left and right sides of the equation ✓ to get that the inverse of 14 is -1 x= 154 ×(y- x(y- 4 Step 3. Multiply both the left and right side of the equation by 154 ×(y- x(y- 4 So the solved form for x is: (mod 51) x (y- 4 ♦ ✔ is the first number that is a multiple of 14 and is equivalent to 1 in mod 51. ✔in mod 51. X in mod 51. = 154 = x (mod 51) ) (mod 51) Solving a Linear Mod Congruence X x 14x (mod 51) by using the property of multiplicative inverses in mod 51
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Solve for :
y = 14x + 4 (mod 51)
Step 1. subtract 4
y- 4
154
Step 2. Find the multiplicative inverse i of 14
We need to solve for in the following congruence:
14. i 1✔ (mod 51)
Consider:
1 = 52 = 103 = 154 = 205 = 256 (mod 51)
✔ = 14x + 4- 4
Now we can solve for ¿ in the regular equation:
14 i 154
154
154
✔from both the left and right sides of the equation
✓
to get that the inverse of 14 is -1
x= 154
×(y-
x(y- 4
Step 3. Multiply both the left and right side of the equation by 154
×(y-
x(y- 4
So the solved form for x is:
(mod 51)
x (y- 4
♦ ✔ is the first number that is a multiple of 14 and is equivalent to 1 in mod 51.
✔in mod 51.
X in mod 51.
= 154
= x (mod 51)
) (mod 51)
Solving a Linear Mod Congruence
X
x 14x (mod 51)
by using the property of multiplicative inverses in mod 51
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