JEE Main 2022 — Binomial Theorem Question with Solution
From: JEE Main 2022 (Online) 26th June Morning Shift
Question
The remainder when (2021)2023 is divided by 7 is :
Choose an option
Show full solutionCorrect option: C
Step-by-step explanation
= (2016 + 5)2023 [here 2016 is divisible by 7]
= 2023C0 (2016)2023 + .......... + 2023C2022 (2016) (5)2022 + 2023C2023 (5)2023
= 2016 [2023C0 . (2016)2022 + ....... + 2023C2022 . (5)2022] + (5)2023
= 2016 + (5)2023
= 7 288 + (5)2023
= 7K + (5)2023 ...... (1)
Now, (5)2023
= (5)2022 . 5
= (53)674 . 5
= (125)674 . 5
= (126 1)674 . 5
= 5[674C0 (126)674 + ......... 674C673 (126) + 674C674]
= 5 126 [674C0(126)673 + ....... 674C673] + 5
= 5 . 7 . 18 [674C0(126)673 + ....... 674C673] + 5
= 7 + 5
Replacing (5)2023 in equation (1) with 7 + 5, we get,
(2021)2023 = 7K + 7 + 5
= 7(K + ) + 5
Remainer = 5
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