remainder

[capnx]

This is a question from a Chinese GMAT practice sample:

"n" is a natural number. When "n" is divided by 3 the remainder is 2 and when divided by 4 the remainder is 1. What is the remainder when n is divided by 12?

a 1
b 2
c 3
d 4
e 5

What'd be the fastest way of doing this? please be clear with the steps (I always get confused with remainders)

The explanation given was really confusing to me:

[spoiler]n = 3a+2 (a is some natural number)
n = 4b+1 (b is some natural number)
so 3a+2 = 4b+1
so 3a+2 = 4*(b'+1) +1 (where b' is some natural number)
3a+2 = 4b'+5
3a-3 = 4b'
3(a-1) = 4b'
so 4b' is a multiple of 4 and a multiple of 3, so 4b' is a multiple of 12
so (4b'+5)/12 => 4b'/12 + 5/12
so 4b'/12 remainder is 0, 5/12 remainder is 5
so e: remainder is 5

What I don't get is the line: 3a+2 = 4*(b'+1) +1
and why 4*(b'+1) +1, why not 4*(b'+2) +1, or +3 +4 +5...[/spoiler]

[Brent@GMATPrepNow]

This is a question from a Chinese GMAT practice sample:

"n" is a natural number. When "n" is divided by 3 the remainder is 2 and when divided by 4 the remainder is 1. What is the remainder when n is divided by 12?

a 1
b 2
c 3
d 4
e 5

What'd be the fastest way of doing this? please be clear with the steps (I always get confused with remainders)

The explanation given was really confusing to me:

[spoiler]n = 3a+2 (a is some natural number)
n = 4b+1 (b is some natural number)
so 3a+2 = 4b+1
so 3a+2 = 4*(b'+1) +1 (where b' is some natural number)
3a+2 = 4b'+5
3a-3 = 4b'
3(a-1) = 4b'
so 4b' is a multiple of 4 and a multiple of 3, so 4b' is a multiple of 12
so (4b'+5)/12 => 4b'/12 + 5/12
so 4b'/12 remainder is 0, 5/12 remainder is 5
so e: remainder is 5

What I don't get is the line: 3a+2 = 4*(b'+1) +1
and why 4*(b'+1) +1, why not 4*(b'+2) +1, or +3 +4 +5...[/spoiler]

Here's the faster way.
Notice that there are no answer choices such as "cannot be determined," which means that there is only one answer to this question.
So, find a number that satisfies both conditions and then see what the remainder is when you divide that number by 12.
Condition 1: When n is divided by 3 the remainder is 2. So, n could equal 2, 5, 8, 11, 14, 17, etc
Condition 2: When n is divided by 4 the remainder is 1. So, n could equal 1, 5, 9, 13, 17, etc
Of course, we would have stopped at 5 here since we see that 5 satisfies both conditions.

So, now check the remainder when we divide 5 by 12.
The remainder is 5 so choose E.

Also notice that 17 also satisfies both conditions. When we divide 17 by 12, the remainder is still 5