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.
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