Problem 5 Solve for x and y xy 5 mod 2801 xy 1 mod 2801 Pl

Problem 5. Solve for x and y.
x+y 5 ( mod 2801)

xy 1 ( mod 2801)

Please, provide step by step worked out solution. Thank you!

Note that 2801 is a prime number.

x+y-5=0(mod 2801)

Hence, x+y-5 is multiple of 2801 ie

x+y-5=2801m

x-y-1=0 (mod 2801)

Hence, x-y-1 is multiple of 2801 ie

x-y-1 =2801n

Adding the two equations gives:

2x-6=0(mod 2801)

2802 =2801+1=1(mod 2801)

2802=1401*2

And since 2801 is prime number we can multiply last equation by 1401

So we get:

1401(2x-6)=0(mod 2801)

1401*2x-1401*6=0(mod 2801)

2801x+x-2802*3=0(mod 2801)

x-(2801+1)*3=0(mod 2801)

x-3=0(mod 2801)

Hence, x=3(mod 2801)

Hence, x=2801k+3, where k is an integer

x+y=5 (mod 2801)

Substituting solution for x gives:

3+y=5(mod 2801)

Hence, y=2(mod 2801)

Hence, y=2801p+2, p is an integer