<p>Hello! I really appreciate your time in advance.
This problem is from Barrons 2400.
On pg. 244, the question asks the following</p>
<p>If n^2 is divisible by k, the n is divisible by k, where n and k are positive inegers. When is this statement true?</p>
<p>I. Whenever k is even
II. Whenever k is odd
III. Whenever k is prime</p>
<p>A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III</p>
<p>Ans. C.</p>
<p>I looked at the solution, but I did not quite get it. Could someone please explain why the answer is C clearly
possibly with an example?
Thank you very much for your help!!</p>
<p>6^2 = 36 which is divisible by 9 but 6 is not.</p>
<p>15^2 = 225 which is divisible by 25 but 15 is not.</p>
<p>For both of these counter-examples, I chose k so that it had two repeating prime factors, only one of which shows up in the n. But if k is prime and it divides into n^2, it will also divide into n. It’s just that n^2 will have that factor appearing twice as many times. To see this, pick some examples that work and look at the prime factorizations.</p>