Tough and Tricky questions: Number Properties.
What is the smallest positive integer x such that 1512x is the cube of a positive integer?  Choices:
A) 7
B) 14
C) 28
D) 35
E) 49
We want to know the smallest x that will make 1512x 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 450 can be visualized:   .........1512   ......../.......\   ......216......7   ...../.. \....   ...36.... 6...   .../. \..../.\..   ..6….6…2…3   ./.\ ../\   2..3 2…3   So, we have 2 * 2 * 2 * 3 * 3 *3 * 7 that can be multiplied together to get 1512. Now we need to figure out what we need to make 1512 * x into a cube of y (y^3=1512*x).   We have two 5s, two 3s, and one 2. To arrange these numbers in identical triples (2,3,7), we need at least one more 5, one 3, and two 2's. Each of these triples will give us the value of y (2*3*7=42), which, multiplied by itself three times, gives us 1512 * x.   Looking at the factors we need to complete the triples, we get   7 * 7= 
<gadget id="calculator">  7 * 7</gadget>
<output>49</output>
49. 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 D.   If time permits, we can do a sanity check. We calculated that y should be 2 * 3 * 7, or 42. 42 * 42 * 42= 74088. Also, 1512 * 49 = 74088.   Answer: 49
<result>E</result>
Tough and Tricky questions: Number Properties.
What is the smallest positive integer x such that 1512x is the cube of a positive integer?