Priyaranjan wrote:What is the greatest common divisor of positive integers j & k.

1) The greatest common divisor of 3j & 2k is 2.
2) The greatest common divisor of 5j & k is 10.

Statement 1: The greatest common divisor of 3j and 2k is 2.
Case 1: j=2 and k=1
Here, 3j = 3*2 = 6 and 2k = 2*1 = 2, with the result that the GCD of 3j and 2k is 2.
In this case, the GCD of j and k is 1.

Case 2: j=2 and k=2
Here, 3j = 3*2 = 6 and 2k = 2*2 = 4, with the result that the GCD of 3j and 2k is 2.
In this case, the GCD of j and k is 2.

Since the GCD of j and k can be different values, INSUFFICIENT.

Statement 2: The greatest common divisor of 5j and k is 10.
Case 3: j=2 and k=10
Here, 5j = 5*2 = 10, with the result that the GCD of 5j and k is 10.
In this case, the GCD of j and k is 2.

Case 4: j=10 and k=10
Here, 5j = 5*10 = 50, with the result that the GCD of 5j and k is 10.
In this case, the GCD of j and k is 10.

Since the GCD of j and k can be different values, INSUFFICIENT.

Statements combined:
Case 3 satisfies both statements.
In Case 3, the GCD of j and k is 2.

To satisfy both statements, j must be EVEN and k must be a multiple of 10.
Thus, the GCD of j and k cannot be less than 2.
Test whether the GCD can be greater than 2.

Case 5: j = 2*2 = 4 and k = 10*2 = 20
Here, the GCD of j and k is 4.
In this case, 5j = 5*4 = 20, with the result that the GCD of 5j and k is 20, violating the constraint in statement 2 that the GCD of 5j and k is 10.
Thus, the GCD of j and k cannot be 4.

Case 6: j = 2*3 = 6 and k = 10*3 = 30
Here, the GCD of j and k is 6.
In this case, 5j = 5*6 = 30, with the result that the GCD of 5j and k is 30, violating the constraint in statement 2 that the GCD of 5j and k is 10.
Thus, the GCD of j and k cannot be 6.

Cases 5 and 6 illustrate that -- if the GCD of j and k is greater than 2 -- then statement 2 cannot be satisfied.
Thus, the GCD of j and k must be 2.
SUFFICIENT.

The correct answer is C.

Priyaranjan wrote:What is the greatest common divisor of positive integers j & k.

1) The greatest common divisor of 3j & 2k is 2.
2) The greatest common divisor of 5j & k is 10.

Statement 1 tells us that both 3j and 2k are divisible by 2.

However k itself may or may not be divisible by 2. If k is divisible by 2, then the greatest common divisor of j and k is 2. If k is not divisible by 2, then the greatest common divisor is 1.

So Statement 1 is insufficient.

Statement 2 tells us that both 5j and k are divisible by 10.

However, j itself may or may not be divisible by 5 or, by extension, 10. Therefore we can't tell whether the greatest common divisor is 10, or some number smaller than 10.

So Statement 2 is insufficient.

Statement 1 locked in that the greatest common divisor of j and k is 2, but it wasn't clear whether k is divisible by 2.

Statement 2 tells us that k is divisible by 10. Therefore k must be divisible by 2.

So, the combined statements provide information sufficient for determining that the greatest common divisor is 2.

The correct answer is C.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

What is the greatest common divisor of positive integers j & k.

1) The greatest common divisor of 3j & 2k is 2.
2) The greatest common divisor of 5j & k is 10.

In the original condition, there are 2 variables(j,k), which should match with the number of equations. So you need 2 more equations. For 1) 1 equation, for 2) 1 equation, which is likely to make C the answer. In 1) & 2), from GCD(3j,2k)=2, J gets 2 as factor. From GCD(5j,k)=10, j gets 2 as factor and k gets 10 as factor. So, j and k always have 2 as factor and GCD(j,k)=2, which is unique and sufficient. Therefore, the answer is C.


->For cases where we need 2 more equations, such as original conditions with "2 variables", or "3 variables and 1 equation", or "4 variables and 2 equations", we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.