Data Sufficiency

sachin_yadav wrote:If n is an integer, is n/7 an integer?

(1) 3n/7 is an integer.
(2) 5n/7 is an integer.

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)
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Okay, onto the question:

Target question: Is n/7 an integer?

Statement 1: 3n/7 is an integer
If 3n/7 is an integer, then we can also say that 3n is DIVISIBLE by 7.
This means that there's a 7 HIDING in the prime factorization of 3n.
Since there's no 7 HIDING in 3, there must be a 7 HIDING in the prime factorization of n.
If there's a 7 HIDING in the prime factorization of n, then n must be divisible by 7
If n is divisible by 7, then n/7 is DEFINITELY an integer.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 5n/7 is an integer
If 5n/7 is an integer, then we can also say that 5n is DIVISIBLE by 7.
This means that there's a 7 HIDING in the prime factorization of 5n.
Since there's no 7 HIDING in 5, there must be a 7 HIDING in the prime factorization of n.
If there's a 7 HIDING in the prime factorization of n, then n must be divisible by 7
If n is divisible by 7, then n/7 is DEFINITELY an integer.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent

I'd recommend that you compare this question with OG #135, which tests a similar principle.

In this problem, your second interpretation is much better: "is n/7 an integer -> is n divisible by 7." It's generally better to approach divisibility problems in conceptual terms rather than algebraic ones.

If in this problem one of the statements had read "14n/7 is an integer," then the statement would be INSUFFICIENT. Since 14 itself is divisible by 7, n could be anything and the statement would still be true.

Since 3 and 5 are not divisible by 7, then if 3n and 5n are each divisible by 7, n must be divisible by 7 - the 3 and 5 can't be helping to divide 7, as 14 was in the previous example.

If in a case like this the coefficient in the numerator shares no factors with the denominator, then the variable must be divisible by the denominator.