What is the highest positive integer less than 1800 that is divisible

-----ASIDE---------------------
A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)
-----ONTO THE QUESTION!---------------------

Let N = the number in question

N is divisible by 2.
So the prime factorization of N looks like this: N = (2)(?)(?)(?)(?)(?)(?)
aside: The extra (?)'s represent other possible prime numbers that COULD be in the prime factorization of N

N is divisible by 3.
So the prime factorization of N now looks like this: N = (2)(3)(?)(?)(?)(?)(?)

N is divisible by 4.
This means there are two 2's in the prime factorization of N
We already have one 2 in our existing prime factorization, so we need only add one more 2 to get: N = (2)(3)(2)(?)(?)(?)(?)

N is divisible by 5.
We get: N = (2)(3)(2)(5)(?)(?)

N is divisible by 6.
6 = (2)(3)
Since we already have (2) and (3) in our prime factorization of N, we need not add more.
We have: N = (2)(3)(2)(?)(?)(?)(?)

N is divisible by 7.
We get: N = (2)(3)(2)(5)(7)(?)(?)(?)

(2)(3)(2)(5)(7) = 420
So, N is some multiple of 420

We are told that N is less than 1800

(420)(4) = 1680 (less than 1800)
(420)(5) = 2100 (too big)

Answer: 1680 (D)

Cheers,
Brent