Data Sufficiency

Mo2men wrote:Is the positive integer x divisible by 21?

(1) When X is divided by 14, the remainder is 4
(2) When X is divided by 15, the remainder is 5

NICE QUESTION!

Target question: Is the positive integer x divisible by 21?

Statement 1: When X is divided by 14, the remainder is 4
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

In this statement, we aren't told how many times 14 divides into x, but we can just assume that 14 divides into x k times. In other words, x divided by 14 equals k with remainder 4"
So, we can write: x = 14k + 4 (where k is some positive integer)
This is enough information to show that x is definitely NOT divisible by 21
How do I know this?

Take x = 14k + 4 and rewrite 14 as follows: x = (7)(2)k + 4
Highlight two values: x = (7)(2)k + 4
Simplify: x = (7)(some integer) + 4
This tells us that x is 4 GREATER than some multiple of 7
In other words, x is NOT divisible by 7.
If x is NOT divisible by 7, then we can be certain that x is NOT divisible by 21.
We know this because, for x to be divisible by 21, x must be divisible by 3 AND x must be divisible by 7
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: When X is divided by 15, the remainder is 5
Let's say that 15 divides into x j times. In other words, x divided by 15 equals j with remainder 5"
So, we can write: x = 15j + 5 (where j is some positive integer)
Rewrite the equation: x = (3)(5)j + 5
Highlight two values: x = (3)(5)k + 5
Simplify: x = (3)(some integer) + 5
This tells us that x is 5 GREATER than some multiple of 3
In other words, x is NOT divisible by 3.
If x is NOT divisible by 3, then we can be certain that x is NOT divisible by 21.
We know this because, for x to be divisible by 21, x must be divisible by 3 AND x must be divisible by 7
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent