50th root of a number x is 12 and 50th root of another number y is 18. Then the remainder obtained on dividing (x + y) by 25 is _________ . [2023]

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Solution

Q.
50th root of a number x is 12 and 50th root of another number y is 18. Then the remainder obtained on dividing (x + y) by 25 is _________ . [2023]

Ans.
(23)

Given, $x^{1 / 50} = 12 \Rightarrow x = {(12)}^{50}$

$y^{1 / 50} = 18 \Rightarrow y = {(18)}^{50}$

$∴$ $\frac{x + y}{25} = remainder \Rightarrow \frac{x + y}{25} = \frac{12^{50} + 18^{50}}{25}$

$\frac{12^{50}}{25} + \frac{{(18)}^{50}}{25} = \frac{{(144)}^{25} + {(324)}^{25}}{25} = \frac{{(150 - 6)}^{25} + {(325 - 1)}^{25}}{25}$

$= \frac{25 k - (6^{25} + 1)}{25} = \frac{25 k - [{(5 + 1)}^{25} + 1]}{25} = \frac{25 k_{1} - 2}{25}$

$∴$ $Remainder = 23$

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