Tough and Tricky questions: Number Properties.
What is the smallest positive integer x such that 500x is the cube of a positive integer?
Choose the correct answer:
A) 2
B) 3
C) 5
D) 10
E) 15
We want to know the smallest x that will make 500x 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 500 can be visualized:   ...........500   ......../.......\   ......50.......10   ...../..\....../...\   ...25...2...2.....5   .../..\   ..5....5   So, we have 5 * 5 * 5 * 2 * 2 that can be multiplied together to get 500. Now we need to figure out what we need to make 450 * x into a cube of y (y^3=500*x).   We have two 5s, two 3s, and one 2. To arrange these numbers in identical sets (2,5), we need at least one more 2. Each of these numbers will give us the value of y (2*5=10), which, multiplied by itself three times, gives us 500 * x.   Looking at the factors we need to complete the sets, we get x= 2. 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 A.   If time permits, we can do a sanity check. We calculated that y should be 2 * 5, or 10. 10 * 10 * 10 = 1000. 2lso, 500 * 2 = 1000.
<result>A</result>
Tough and Tricky questions: Number Properties.
What is the smallest positive integer x such that 500x is the cube of a positive integer?