Data Sufficiency

a problem from a student

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Problem -

Is n/18 an Integer ?

1) 5n/18 is an integer
2) 3n/18 is an integer
When the question states is N/18 an integer.

My approach is to use the prime box concept. So in this case N should have atleast 2*3*3 in its prime box if it has to be an integer -

Considering statement one. They say 5n/18 is an integer. Does this not imply that N does have atleast (2*3*3) in its prime box so the outcome is an integer

Similarly for stem 2. I know I am wrong somewhere can you please help and clarify this concept. OA - C and I get D

I see a lot of problems on factors, multiples and integers (number theory mostly) Any good resource to follow

Thanks,
R

ah.

the prime box concept isn't terribly useful for addressing "n" in this problem, because, unfortunately, the problem doesn't actually specify that "n" has to be an integer. therefore, you must consider non-integer values of n when you consider the statements.

statement 1 alone:
if 5n/18 = 5, then n = 18, for which n/18 is an integer; if 5n/18 = 1, then n = 18/5, which is not an integer (and so therefore n/18 isn't an integer either). insufficient.
note that we are picking values for 5n/18, NOT values for n - per the problem statement!

statement 2 alone: reduce 3n/18 to n/6 to make this statement easier to think about.
if n/6 = 3, then n = 18, for which n/18 is an integer; if n/6 = 1, then n = 6, for which n/18 is not an integer. insufficient.
note that we are picking values for 3n/18, NOT values for n - per the problem statement!)

together:
remember that sums and differences of integers are also integers.
if 5n/18 and 3n/18 are integers, then 5n/18 - 3n/18 = 2n/18 is also an integer. once we have that, 3n/18 - 2n/18 = n/18 is also an integer. sufficient.

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if the problem had stated that n must be an integer, then statement (1) would become sufficient, but statement (2) would still be insufficient.