Suppose that x is an integer that satisfies the following congruences:

3 + x == 2^2 (mod 3^3)

5 + x == 3^2 (mod 5^3)

7 + x == 4^2 (mod 7^3)

What is the remainder when x is divided by 105?

wiseowl Jan 30, 2022

#1**+1 **

I've done this one before somewhere ... so has Heureka.

I'll start you off (although there may be an easier way)

3 + x == 2^2 (mod 3^3)

3 + x == 4 + 3^3K

x == 1 + 3^3K

this would also mean that

**x == 1 + 3K**

5 + x == 3^2 (mod 5^3)

5 + x == 9 + 5^5L

x == 4 + 5^3L

x == 4+ 5L

**x == -1 + 5L**

7 + x == 4^2 (mod 7^3)

7 + x == 16 + 7^3M

x == 9 + 7M

**x == 2 +7M**

First I solved 1+3K = -1+5L and got x= -1 mod 15

Then I solved x=1+15a with x = 2+7M and got x=79+105b

So I got an answer of 79 but I could easily have made a careless error.

I use the Euclidean Algorithm to solve the diophantine equations.

Melody Jan 31, 2022