Question

Does the congruence equation

6x ≡ 7(mod 25)

have a solution for x (notice that congruence only makes sense if x ∈ Z so you are looking for integer

solutions)? If it does, find the solution. If it does not, prove that it does not.​

Answer 1

Answer:

x = 22.

Step-by-step explanation:

6x ≡ 7(mod 25)

6x ≡ 25n + 7  where x and n are integers.

Checking multiples of 25:

25:  (n= 1)   25+7 = 32. 6*5 = 2 (mod 25) . Does not fit.

50, 75 , 100  do not fit either.

The answer  is 22 because 6*22 = 132 =  125 + 7 = 7(mod 25).

Answer 2

The solution to the congruence equation  6x ≡ 7(mod 25)  is x = 22.

 

We have the congruence equation

6x ≡ 7(mod 25)

6x ≡ 25n + 7  where x and n are integers.

So, it means 25 divides 6x-7.

As 25 divides 6x-7 . So value of x should be greater than 5.

It does not work for 6,7,..... 21.  

For x=22

$6*22 = 132 \\= 125 + 7 \\= 7(mod 25).$  

The solution for the congruence equation 6x ≡ 7(mod 25) is x=22.

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