Question

a SO
possess
4. Find the least positive incongruent solutions of:
(a) 13x = 9 (mod 25)
(b) 259x = 5 (mod 11)​

Answer 1

The least positive incongruent solutions of (a) 13x = 9 (mod 25) is x == 18(mod 25). and (b) 259x = 5 (mod 11) is x == 10(mod 11)

Given,

13x = 9 (mod 25)

259x = 5 (mod 11)

To Find,

Find the least positive incongruent solutions of 13x = 9 (mod 25) and 259x = 5 (mod 11)

Solution,

(a) 13x = 9 (mod 25)

(13, 25) = 1, the congruence has a single solution

25=13 × 1+I2

13 = 12 ×1 + 1

12 = 12 x 1

1 = 13 - 12 x 1 = 13 - (25 - 13 x 1)

or, 1 = 13 x 2 - 25 x 1

Then which gives,

9 = 13 x 2 x 9 - 25 x I x 9,

9 = 13 x 18 - 25 x 9 == I3x(mod 25)

x == 18(mod 25).

(b) 259x = 5 (mod 11)

259 = 11 x 25 + 6.,

or 259 == 6(mod 11),

:. we get,

So,

6x == 259x == 5(mod 11)

6x == 5(mod 11).

Now the above has only one solution.

We observe that among 0, 1, 2, ... , 10, x = 10 satisfies the above.

so, x == 10(mod 11) is the only solution.

Hence, the least positive incongruent solutions of (a) 13x = 9 (mod 25) is x == 18(mod 25). and (b) 259x = 5 (mod 11) is x == 10(mod 11)

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