Concept: The question tests on Number Properties(Factors)
This question is a great candidate for the “Break down the question statement” strategy.
Positive integer k has two positive prime factors, 3 and 7. This means that k is definitely a composite number since it has more than 2 factors (remember that 1 and k are already factors of k, so there are more than 2 factors for sure).
For any composite number N which can be written as N = ax∗by∗czax∗by∗cz* ……………, the number of positive factors of N is given by the expression (x+1) (y+1) (z+1)……….
As per the question statement, k can be written as k = 3x∗7y3x∗7y. Therefore, the number of factors for k = (x+1) (y+1). The questions tells us that k has a total of 6 positive factors.
So, (x+1) (y+1) = 6.
Note that x and y are positive integers, therefore, it’s easier to deal with the equation by plugging in values for x and y than multiplying the terms in the bracket.
If (x+1) = 3, (y+1) = 2; so x = 2 and y = 1 and k = 32∗7132∗71.
If (x+1) = 2, (y+1) = 3; so x = 1 and y = 2 and k = 31∗7231∗72.
By breaking down the question, we understand that there can be only two possible values for k. Let’s now use the statements to figure out which one it is.
From statement I alone, 3232 is a factor of k. This means k = 32∗7132∗71, since a factor cannot be bigger than the number it divides (3232 cannot divide 3131)
Statement I alone is sufficient. Answer options B, C and E can be eliminated. Possible answer options are A or D.
From statement II alone, 7272 is not a factor of k. This again means k = 32∗7132∗71.
Statement II alone is sufficient. Answer option A can be eliminated.
The correct answer option is D.