Consider the congruence 6x=15(mod 21). How many solutions does the congruence have? What are all the solutions?
X congruent 6 (mod 7)
To determine the number of solutions and find all the solutions to the given congruence 6x ≡ 15 (mod 21), we can start by reducing the congruence to its simplest form. In this case, we need to divide both sides of the congruence by the greatest common divisor (GCD) of 6 and 21, which is 3.
Dividing both sides by 3, we get 2x ≡ 5 (mod 7). Now let's find the multiplicative inverse of 2 modulo 7. The multiplicative inverse of a modulo m is a number x such that ax ≡ 1 (mod m). In this case, we need to find x such that 2x ≡ 1 (mod 7).
To find the inverse, we can try out different values until we find a value that satisfies the equation. Trying x = 1, 2, 3, etc., we find that x = 4 is the multiplicative inverse of 2 modulo 7, because 2 * 4 ≡ 1 (mod 7).
Now, multiply both sides of the congruence by the multiplicative inverse, which is 4 in this case:
2x * 4 ≡ 5 * 4 (mod 7)
8x ≡ 20 (mod 7)
Simplifying further:
x ≡ 20 ≡ 6 (mod 7)
Therefore, the congruence 6x ≡ 15 (mod 21) has a unique solution. The solution to the congruence is x ≡ 6 (mod 7).