Bars of Soap

Relevant wiki: Number of Factors

From the first sentence, we know that when 2007 is divided by $N$ , the remainder is 5.

Or equivalently, $2007 \equiv 0 5 \pmod N$ , or $2007 - 5 = 2002 \equiv 0 \pmod N$ .

So 2002 must be divisible by $N$ , with and additional constraint that $N$ is larger than 5.

And we want to find the total number of possible values of $N$ , which is simply the number of positive divisors of 2002 larger than 5.

2002 factors to $2^1\times7^1\times11^1\times13^1$ . So total number of factors of 2002 is $(1+1)(1+1)(1+1)(1+1) = 16$ .

But the only positive divisors of 2002 less than or equal to 5 are (1,2), so we need to discount two of these divisors.

Hence, there are $16-2 = \boxed{14}$ possible choices of $N$ .

2007-5=2002. What ever factors we get is the number N. We may use number theory or simple logic as under.
Factors of 2002 are 2, 7, 11, 13.
Multiplying one at a time we have 4 box arrangments........Bars=..2..7..11..13.....................N=....1001.....286....182....154
Multiplying two at a time we have 6 box arrangments........Bars=..14..22..26..77..91..143 ....N=..143..91..77...26..22..14
Multiplying three at a time we have 4 box arrangments. But with 7 11 13=1001 boxes there will be only two in a box and we can not have 5 remainder. Hence we get only 3 box arrangments.............Bars=..154..182..286............... ...N=...13...11...7
Multiplying four at a time we have 1 box arrangment.........Bars=...2002...............................N=..1..
Total 4+6+3+1=14.

C(4,1)= 4 .......C(4,2)= 6 ......C(4,3)=4...minus one as explained above= 3 .......C(4,4)= 1 .