As Exponents (Part 3)

Algebra Level 2

What is x y xy if x x and y y are positive integers?

{ x x + y y = 46912 x y + y x = 5392 \large \begin{cases} x^{x}+y^{y}=46912 \\ x^{y}+y^{x}=5392 \end{cases}


The answer is 24.

Kaizen Cyrus by Kaizen Cyrus , 18, Philippines

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1 solution

Henry U
Feb 24, 2019

Since 7 7 = 823543 > 46912 7^7 = 823543 > 46912 , WLOG x ≤ 6 x \leq 6 .

We can try x = 6 x = 6 and get that y y = 46912 − 6 6 = 256 ⇔ y = 4 y^y = 46912 - 6^6 = 256 \Leftrightarrow y=4

Then, x y + y x = 6 4 + 4 6 = 5392 x^y + y^x = 6^4 + 4^6 = 5392 which fits with the second equation.

Therefore, x y = 6 ⋅ 4 = 24 xy = 6 \cdot 4 = \boxed{24} .

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