KVPY 2020-21 SA Exam Previous Papers

Question : 7 of 80

Marks: +1, -0

The number of ordered pairs (x, y) of positive integers satisfying

2^x + 3^y = 5^\xy

Solution:

∵

2^x + 3^y = 5^\xy

When x = y = 1 then 2 + 3 = 5
and

({2}/{5})^{x} \.{1}/{5^{y}}+({3}/{5})^{y} \.{1}/{5^{x}}=1

here for any

x, y \∈ Z^{+}

L.H.S. is not equal to 1 as both are less than

1/2 .

∴ Only one ordered pair (1, 1) is possible.

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