How do you solve x 2 congruent to a mod p?

How do you solve x 2 congruent to a mod p?

The congruence x2 ≡ a (mod p) either has no solutions or two solutions. If x is a solution, so is –x. Euler’s Criterion says that an odd integer a relatively prime to p is a quadratic residue (mod p) if and only if a(p-1)/2 ≡ 1 (mod p).

How do you know if a mod is congruent?

A simple consequence is this: Any number is congruent mod n to its remainder when divided by n. For if a = nq + r, the above result shows that a ≡ r mod n. Thus for example, 23 ≡ 2 mod 7 and 103 ≡ 3 mod 10. For this reason, the remainder of a number a when divided by n is called a mod n.

How do you find all solutions of congruence?

Generally, a linear congruence is a problem of finding an integer x that satisfies the equation ax = b (mod m). Thus, a linear congruence is a congruence in the form of ax = b (mod m), where x is an unknown integer. In a linear congruence where x0 is the solution, all the integers x1 are x1 = x0 (mod m).

What are incongruent solutions?

6. Incongruent (in this case) means distinct modulo 1562. For example, 1 and 1561 are incongruent modulo 1562, but 1 and 1563 are not (rather, they are congruent modulo 1562).

What is mod p?

Definition. ( i) The “mod p” numbers are all the remainders: {0,1,2,…,p − 1} when a natural number is divided by p.

Is modulo 3 congruent?

We say integers a and b are “congruent modulo n” if their difference is a multiple of n. For example, 17 and 5 are congruent modulo 3 because 17 – 5 = 12 = 4⋅3, and 184 and 51 are congruent modulo 19 since 184 – 51 = 133 = 7⋅19. The rational numbers 1/2 and 13/2 are congruent modulo 3 because 13/2 – 1/2 = 6 = 2⋅3.

How many solutions does the equation 6x 4 have in Z7?

2 solutions
(a) If n = 7, then d = (6, 7) = 1 and 1 | 4, so that the equation 6x = 4 has d = 1 solution in Z7. (b) If n = 8, then d = (6, 8) = 2 and 2 | 4, so that the equation 6x = 4 has d = 2 solutions in Z8.