Suvorov wrote:

Bunuel wrote:

only one such pair is possible x=4>y=2: x^y=4^2=16=2^4=y^x --> x^4-y^4=240. Sufficient.

How do you come to this conclusion? Just picking numbers/knowing or is there some mathematical rule for this? I understand that it revolves around the power of 2, but can't put my finger on it.

I think it's worth remembering that

4^2=16=2^4, I've seen several GMAT questions on number properties using this (another useful property

8^2=4^3=2^6=64).

But if you don't know this property:

Given:

x^y = y^x and

x>y, also

x and

y are

distinct positive integers.

Couple of things:

x and

y must be either distinct positive odd integers or distinct positive even integers (as odd in ANY positive integer power is odd and even in ANY positive integer power is even).

After testing several options you'll see that

x=4 and

y=2 is the only possible scenario: because, when

y\geq{2} and

x>{4}, then

x^y (bigger value in smaller power) will be always less than

y^x (smaller value in bigger power):

5^3<3^5, or

6^2<2^6, or

8^2<2^8, or

10^2<2^{10}, ....

_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:

PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.

What are GMAT Club Tests?

25 extra-hard Quant Tests