Remainders questions help If we divide a number by 3, 4 ,5 , 6 , we have  the remainders 2, 3 , 4 , 5. Is there any way to  get a pattern without guessing so many numbers and checking by 3, 4 ,5 ,6?
Hint: We are told that our number $x$ is congruent to $-1$ modulo $3$, $4$, $5$, and $6$.
This is first Modular Arithmetic problem I'm solving, I hope it's correct.
We have
$$-1\equiv3 \mod( y )$$
$$-1\equiv4\mod (y)$$
$$-1\equiv5\mod (y)$$
$$-1\equiv6\mod (y)$$

then
$$-1\equiv60\mod(y)$$

Any Number of the form $60p-1$, ( $p \in \mathbb N$) will have mentioned properties.