The integer K is positive, but less than 400. If 21K is a multiple of 180, how many unique prime factors does K have?  Choose one.
A) 1 B) 2 C) 3 D) 4 E) 5
Also note here the relevance of 'K must be less than 400'.   21K is 180n (a multiple of 180).   180n = 2^2 * 3^2 * 5 * n = 3*7*K   n must have a 7 at least.   K must have two 2s, a 3 and a 5 at least. This means it must be at least   2*2*3*5= 
<gadget id="calculator">  2*2*3*5</gadget>
<output>60</output>
60.   So K has the following prime factors: 2, 3 and 5. Can it have any other prime factors? The next smallest prime factor is 7. But   60*7= 
<gadget id="calculator">  60*7</gadget>
<output>420</output>
420 - a number greater than 400. This means that if K is greater than 60, the only other prime factors that K can have must be out of 2, 3 and 5 only. That is, K may be 60*2 or 60*3*2 or 60*5 etc. This tells us that K has exactly 3 prime factors. If we did not have this condition of K less than 400, we would not know exactly how many factors K has.   Answer (3)
<result>C</result>
The integer K is positive, but less than 400. If 21K is a multiple of 180, how many unique prime factors does K have?