Suppose we are told that:
Positive integer $n$ leaves a remainder of 4 after division by 6 and a remainder of 2 after division by 8. What is the remainder that n leaves after division by 12?
The statement "positive integer $n$ leaves a remainder of 4 after division by 6" can be expressed as: $n=6p+4$. Thus according to this particular statement nn could take the following values: $4, 10, 16, 22, 28, 34, 40, 46, 52, 58, 64, ...$
The statement "positive integer n leaves a remainder of 2 after division by 8" can be expressed as: $n=8q+2$. Thus according to this particular statement nn could take the following values: $2, 10, 18, 26, 34, 42, 50, 58, 66, ...$
The above two statements are both true, which means that the only valid values of nn are the values which are common in both patterns. For example nn can not be 16 (from first pattern) as the second formula does not give us 16 for any value of integer qq.
So we should derive general formula (based on both statements) that will give us only valid values of $n$.
How can these two statement be expressed in one formula of a type $n=kx+r$? Where xx is divisor and rr is a remainder.
Divisor $x$ would be the least common multiple of above two divisors 6 and 8, hence $x=24$.
Remainder rr would be the first common integer in above two patterns, hence $r=10$.
Therefore general formula based on both statements is $n=24k+10$. Thus according to this general formula valid values of $n$ are: $10, 34, 58, ...$
Now, $n$ divided by 12 will give us the reminder of 10 (as 24k is divisible by 12).