Find the solutions of the congruence 15x^2 + 19x = 5 (mod 11). [Hint: Show the c

Question

Find the solutions of the congruence 15x^2 + 19x = 5 (mod 11). [Hint: Show the congruence is equivalent to the congruence 15x^2 + 19x + 6 = 0 (mod 11). Factor the left-hand side of the congruence; show that a solution of the quadratic congruence is a solution of on eof hte two different linear congruences.]

It was easy to follow the hint of adding 6 to both sides and factoring the left-hand side, but to solve the congruences, I didn't know how. The solution manual says that the inverse of 5 modulus 11 is 9, and multiplying both sides of 5x + 3 = 0 (mod 11) by 9 yeilds x + 27 = 0 (mod 11), so x = -27 = 6 (mod 11). Similarly, an inverse of 3 modulo 11 is 4, and we get x = -8 = 3 (mod 11). So the solution set is {3, 6} (and anything congruenct to these modulo 11).

I don't understand how they got the inverse for both factors and why mutliplying by each inverse to its factor gets you to your x's. Can someone please show how they did this and explain why it is like that? Thanks

Explanation / Answer

5 * 9 = 45, which equals 1 mod 11.

3 * 4 = 12, which equals 1 mod 11.

This can just be done by trial and error. You only have 9 values to check (11, but no reason to check 0 and 1).

The equation is (5x+3)(3x+2) = 0, as you state.

Then, 5x = -3 is the solution.

If we have 5x = 1, the inverse, we get get

5(9) = 1

Then, we simply multiply both sides by -3 to get 5x=-3mod 11

5(9*-3) = -3 mod 11

9 * -3 = -27

-27 = -33 + 6 = 6 mod 11, so the solution is 6.

Similarly, from 3x+2 = 0, we have

3x = -2

We have the solution 3(4) = 1 mod 11, so, multiplying both sides by -2

3(4*-2) = -2 mod 11

4*-2 = -8

-8 = -11+ 3 = 3 mod 11

The solution is 3

Thus, the solutions are 6 and 3.

We may plug in and verify.

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.