25 190 - 19 190 - 8 190 + 2 190 is divisible by [2023]

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Solution

Q.
$25^{190} - 19^{190} - 8^{190} + 2^{190}$ is divisible by [2023]

1 both 14 and 34

2 34 but not by 14

3 14 but not by 34

4 neither 14 nor 34

Ans.
(2)

Given, $25^{190} - 19^{190} - 8^{190} + 2^{190}$

So, ${(17 + 8)}^{190} - 8^{190} + {(19 - 17)}^{190} - 19^{190}$

$= {}^{190}{C_{0}} 17^{190} + {}^{190}{C_{1}} 17^{189} \times 8 + \dots + 8^{190} - 8^{190} + {}^{190}{C_{0}} 19^{190} - {}^{190}{C_{1}} 19^{189} \times 17 + \dots + 17^{190} - 19^{190}$

$= 17 (2 K_{1}) - 17 (2 K_{2}), where K_{1} and K_{2} are integers.$

So, given number is divisible by 34 but not by 14.

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