In exercise 8-13, solve the system of congruences
The system of congruence is obtained as .
Let be pair wise relatively prime positive integers, (it means that ) . Assume that the , are any integers.
Then the system,
has a solution.
Assume that the numbers m and n are relatively prime integers. Then the following system of congruence equations has a solution.
Now from the above two equation solution can be obtained as,
Here the numbers u and v are found such that the following equation is satisfied.
mu + nv = 1 …… (4)
Now the given congruence equations are:
Compare the equations (5) and (6) with (1) and (2) to obtain the following values,
m = 5 , n = 6 , a = 2 and b = 0
Now find out the values of u and v such that the equation (4) is satisfied.
5u +6v = 1
Above equation will get satisfied with the values u = 1 and v = -1
Now substitute 0 for b , 5 for m, -1 for u, 2 for a, 6 for n and 1 for v into the equation (3)
So the required solution is given by,
Now the third system of congruence is given by,
Again compare the equations (7) and (8) with (1) and (2) to obtain the following values,
Substitute 30 for m and 7 for n into the above equation, and again find the values of u and v such that the equation (4) is satisfied.
From the heat and trial method, above equation holds true for u = -3 and v = 13
Now substitute 3 for b, 30 for m, -3 for u, 12 for a, 7 for n and 13 for v into the equation (3)
Now from the equation (7), (8) and (9), the required solution can be calculated,
Therefore the final solution is .
Let be given by ,where is the congruence class of in . The function may be thought of as representing t as an element of role="math" localid="1658833286608" by taking its least residues.
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