Find at least three numbers that satisfy all three conditions:
(1) there is a remainder of $1$ when the number is divided by $2$;
(2) there is a remainder of $2$ when the number is divided by $3$;
(3) there is a remainder of $3$ when the number is divided by $4$.
Since the LCM($2, 3, 4$) is $12$, all of them can divide $12$ without giving a remainder.
So, $12 - 1=11$ gives
a remainder of $1$ when divided by $2$,
a remainder of $2$ when divided by $3$,
a remainder of $3$ when divided by $4$.
Similarly $12*2-1=23$
and $12*3-1=35$
How would I explain to middle school students the "pulling out of a hat" of
first the $LCM-1$,
and secondly the $LCM*(n)-1$ parts?