determine whether each congruence is solvable if the congrue

determine whether each congruence is solvable. if the congruence is solvable, find the number of incongruent solutions.

Solution

17 is a prime number:

a and 17-a give the same solution for:x^4 mod 17

So we only consider :x=0,1,2,3,4,5,6,7,8 to find solutions to :

x^4=12 mod 17

Note that if x is a solution to x^4 =12 mod 17 then, x^2 is a solution to:x^2=12 mod 17

So we first look at solutions to :x^2 =12 mod 17

x=0 is not a solution.

x=1 is not a solution

x=2    ,2^2 is not equal to 12 mod 17

x=3    ,3^2 is not equal to 12 mod 17

x=4    ,4^2=16 is not equal to 12 mod 17

x=5    ,5^2=25=8 is not equal to 12 mod 17

x=6, 6^2=36=2 mod 17

x=7, 7^2=49=15 mod 17

x=8, 8^2=64=13 mod 17

So for no x is :x^2=12 mod 17 a solution.

Hence, x^4=12 mod 17 has no solutions.

Let x be a solution to:x^6=7 mod 19

Then, x^3 is a solution to y^2 =7 mod 19

So , 7 must be a remainder modulo 19

Let us look at remainders of squares modulo 19

x=0, gives 0

x=1, gives 1

x=2, gives 4

x=3, gives 9

x=4, gives 16=7

x=5, gives 25=6

x=6, gives 36=-2

x=7, gives 49=11

x=8, gives 64=7

x=9, gives 81=5

No need to compute any further as :a and 19-a give the same remainder.

So we see:x=8 and x=4 give remainder 7 mod 19

ie they are solutions to :y^2=7 mod 19

Now if we find: z so that: z^3=8 or 4 mod 19

Then ,(z^3)^2=7 mod 19

So let us look at remainder of cubes modulo 19

x=0 gives 0

x=1 gives 1

x=2 gives 8

x=3 gives 27 =8

x=4 gives 64=7

x=5 gives 125=11

x=6 gives 216=7

x=7 gives 7^2*7=49*7=11*7=1

x=8 gives 8^3=-1

x=9 gives 9^3=7

x=10 gives 10^3=100*10=5*10=50=12

x=11 gives -8^3=1

x=12 gives -7^3=-1

x=13 gives -6^3=-7=12

x=14 gives -5^3=-11=8

x=15 gives: -4^3=-7 =12

x=16 gives:-3^3=-8=11

x=17 gives:-2^3=-8=11

x=18 gives:-1^3=-1

So the solutions are:

x=2,3,14 mod 19