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 whenever ) . 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.
Now the given congruence equations are:
Compare the equations (5) and (6) with (1) and (2) to obtain the following values,
, , and data-custom-editor="chemistry"
Now find out the values of and such that the equation (4) is satisfied.
Above equation will get satisfied with the values and
Now substitute 3 for b , 5 for m , -1 for u , 1 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,
, , and
Substitute 30 for m and 11 for n into the equation (4), 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 and
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,
Now the fourth congruence equation is given by,
Now again compare the equation (10) and (11) with equation (1) and (2) to obtain the following values,
, , and
Substitute 330 for m and 13 for n into the equation (4)
From the heat and trial method and holds true for the above equation,
Now substitute 10 for b , 330 for m , 8 for u , -39 for a , 13 for n and -203 for v into the equation (3)
Now from the equation (10), (11) and (12), the required solution of the congruence is,
Therefore the system of congruence is obtained as .
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