by jlucero Tue Jul 02, 2013 2:42 pm
Algebraically, I'd say you can think of the two statements as this:
1) x = a^3
2) x = b^4
(where a and b are both integers)
Combined, we see that a^3 = b^4, and think about this logically. What numbers could I plug in here? First off, notice that a and b would have to have the same prime factors in order for them to be equal. So when I plug a simple prime number, such as 2, could a=2? No. Because on the other side of this equation, there has to be some multiple of 4 factors of 2. Let's test out other numbers that have two as its only prime factor:
a = 2, a^3 = three 2s
a = 4, a^3 = six 2s
a = 8, a^3 = nine 2s
a = 16, a^3 = twelve 2s
Only when a = 16, would there be some number of 2s that is a multiple of 4. That's why combined, this is enough information- there are twelve 2s, so x is some integer to the twelfth power.
The ultimate short cut looking back at this problem is to realize that when a^3 = b^4, there needs to be the same number of prime factors on both sides of this equation. And since those prime factors need to be able to be grouped into piles of 3 or piles of 4, they must be a multiple of 12.
Joe Lucero
Manhattan GMAT Instructor
