Tough and Tricky questions: Number Properties.
What is the smallest positive integer x such that 324x is the cube of a positive integer?
Choose the correct choice from the following answers:
A) 9
B) 12
C) 18
D) 24
E) 36
We want to know the smallest x that will make 324x a CUBE of some number. Let's call that number y.   Let's first figure out what we're working with. The prime factorization of 324 can be visualized:   ...........324   ......../.......\   ......81.......4   ...../..\....../...\   ...27...3...2.....2   .../..\   ..9....3   /\   .3 3   So, we have 3 * 3 * 3 * 3 * 2 * 2 that can be multiplied together to get 324. Now we need to figure out what we need to make 324 * x into a cube of y (y^3=324*x).   We have four 3s, and two 2s. To arrange these numbers in identical triples (2,3,3), we need at least one more 2 and two 3's. Each of these triples will give us the value of y (2*3*3=18), which, multiplied by itself three times, gives us 324 * x.   Looking at the factors we need to complete the triples, we get   3 * 3 * 2= 
<gadget id="calculator">  3 * 3 * 2</gadget>
<output>18</output>
18. We know this is the smallest number possible because prime factors by definition cannot be broken down any further.   Therefore, we can go with answer choice C.   If time permits, we can do a sanity check. We calculated that y should be 2 * 3 * 3, or 18. 18 * 18 * 18= 5832. Also, 324 * 18 = 5832.   Answer: 18
<result>C</result>
Tough and Tricky questions: Number Properties.
What is the smallest positive integer x such that 324x is the cube of a positive integer?