Bunuel wrote:
If k is a positive integer, is \(\sqrt{k}\) an integer ?
(1) k is a multiple of every single-digit prime number.
(2) The tens digit of k is a factor of a single digit prime number.
The single digit prime numbers are 2, 3, 5, and 7.
Statement 1: K is a multiple of every single digit prime number.
This means that K is a multiple of each of 2, 3, 5, and 7. This means K must a multiple of the LCM of 2, 3, 5, and 7 - which is 210.
Therefore, K is of the form = 210p where p is any positive integer.
Case 1: p = 1 therefore K = 210 which is not a perfect square. Hence, √K is not an integer.
Case 2: p = 210 therefore K = 210*210 = 44100 which is a perfect square. Hence, √K is an integer.
Statement 1, therefore, is insufficient. Eliminate A and D. Statement 2: The tens digit of K is a factor of a single digit prime number.
The single digit prime numbers are 2, 3, 5, and 7. And all possible factors of these are 1, 2, 3, 5, and 7. The tens digit of K can be any of these five integers.
For ex: Let's say the tens digit of K is 2.
Case 1: K = 25, clearly K is a perfect square and √K is an integer.
Case 2: K = 27, clearly K is
not a perfect square and √K is not an integer.
Statement 2, therefore, is insufficient. Eliminate B. Combining the two statements K = 210p and tens digit of K is either 1 or 2 or 3 or 5 or 7.
In order for K to be a perfect square, each of its existing factors must first be squared. Which means p must at least be 210, and then can optionally be multiplied by Z where Z is a non-zero perfect square.
Which means for K to be a perfect square, K = 210*210*Z = 44100*Z where Z is a non-zero perfect square. Multiplying 44100 by any positive integer Z will result in two 0s at the end.
Which means for K to be a perfect square, the tens digit of K ought to be 0 and nothing other than that. However, we've been given from statement 2 that the tens digit of K is either 1 or 2 or 3 or 5 or 7.
Hence, clearly, upon combining the two statements we can say that K is not a perfect square and √K is not an integer. Hence, option C.Posted from my mobile device