If n = 2×3×5×7×11×13×17, then which of the following..

We know that n = 2 * (some huge odd number). From there:

(i) For n to divide by 600, it must divide by 6 * 100, or 6 * 10 * 10, or 6 * 2 * 5 * 2 * 5. But we saw that n only has ONE factor of 2, and to divide by 600 it'd need THREE factors of 2, so (i) is NOT true.

(ii) if n + 19 = 19 * something, then we can say

n + 19 = 19 * something

n = 19 * something - 19

n = 19 * (something - 1)

So would be a multiple of 19, because it equals 19 times some number. But n ISN'T a multiple of 19: its odd factors only go up to 17, and 19 is prime. (ii) is NOT true.

(iii) If (n + 4)/2 = even, then

(n + 4) = 2 * even

n = 2 * even - 4

Now any even number is a multiple of 2, so we can rewrite "even" as "2 * something":

n = 2 * (2 * something) - 4

n = 4 * something - 4

n = 4 * (something - 1)

and this is false for the same reason that (ii) is false, n itself is NOT a multiple of 4.