A bit tough to explain in a reddit comment but I'll try. Since you want 1+x to be divisible by 3, you can write 1+x = 3k ; and 1+4x = 5l.

Now writing x in function of k gives you x=3k-1, and substituing in the second equation gives 12k=5l +3

This means 12k - 3 is divisible by 5. This means the remainder of 12k in its euclidian division by 5 is 3. This sentence is translated, in mathematical terms, as follow : 12k = 3 (mod 5) (the equal sign normally has 3 lines). Here comes the part where you really should try and go through the basic of modular arithmetics in order to truly understand.

The remainder of 12k is exactly the same as the one of 2k, and so you have 2k = 3 (mod 5).

Multiplying by 3 gives 6k = 9 (mod 5), this is k = 4 (mod 5), because 6k and k have the same remainder, and so do 9 and 4.

Thus, every number k which has a remainder 4 in its euclidian division by 5 works : 4 ; 9 ; 14... Those numbers are written 5m + 4. Since x = 3k-1, this means the solutions are the numbers 15m + 11.

Note that 11 is an "obvious" answer, meaning its the first one that you find ; while 15 is the least common multiple of 3 and 5.....