GreenlightTestPrep wrote:
If x, y and z are three different non-negative integers, which of the following COULD be true?
i) \(|x-y|=|x+y|=|y-z|\)
ii) \(x^y = y^z\)
iii) \(x^3 + y^3 = z^3\)
A) i only
B) ii only
C) iii only
D) i and ii
E) i and iii
I created this question to illustrate the importance of checking the answer choices each time you analyze one of the statements. Here’s why:
Once we know that statement i COULD be true (it's true when x = 0, y = 1 and z = 2), we should check the answer choices….
ELIMINATE answer choices B and C, since they incorrectly state that statement i CANNOT be true.
Once we know that statement ii COULD be true (it's true when x = 8, y = 2 and z = 6), check the remaining answer choices….
ELIMINATE answer choices A and E, since they incorrectly state that statement ii CANNOT be true.
So, by the process of elimination, we know that
the correct answer is DBest of all, we’re able to determine the correct answer
WITHOUT analyzing statement iii, which is great since statement iii is VERY TRICKY.
By the way, if you had a hard time determining whether statement iii could be true, you’re not alone!
Statement iii is something mathematicians have struggled with since 1637, when a French mathematician named Pierre de Fermat (shown here)…
…suggested that the equation \(a^3 + b^3 = c^3\) cannot have a solution in which the values of a, b and c are POSITIVE integers.
In fact, Pierre de Fermat went even further to say that the equation \(a^n + b^n = c^n\) cannot have a solution in which the values of a, b and c are POSITIVE integers, for n > 2.
Unfortunately Fermat never proved his theorem. In fact, the theorem (famously known as
Fermat’s Last Theorem) was found after his death. Fermat had written the theorem in the margin of one of his textbooks. In addition to the theorem, Fermat wrote that he had a marvelous proof for it, but that it was too long to fit in the margin of the book.
Since then, pretty much every mathematician has tried to prove (or disprove) Fermat’s Last Theorem, but it took until 1994 (357 years later!!!) when it was successfully proved by a mathematician named Andrew Wiles.
Cheers,
Brent