Verified Solution • 2025–26 Session moderate

QJEEM22Mae8f4: JEE Main 2022 Mathematics - Class 11 JEEM22Mae8f4 Solution: The remainder when is divided by 5 is :

The remainder when 72022+320227^{2022}+3^{2022} is divided by 5 is :

Multiple Choice Options

  • A 0
  • B 2
  • C 3
  • D 4

Step-by-Step Verified Solution

amp;72022+32022amp;=(72)1011+(32)1011amp;=(50−1)1011+(10−1)1011amp;=(501011−1011.501010+…−1)amp;+(101011−1011.101010+…..−1)amp;=5m−1+5n−1=5(m+n)−2amp;=5(m+n)−5+3=5(m+n−1)+3amp;=5k+3amp;∴ Remainder =3\begin{aligned}& 7^{2022}+3^{2022} \\\\& =\left(7^2\right)^{1011}+\left(3^2\right)^{1011} \\\\&=(50-1)^{1011}+(10-1)^{1011} \\\\&= (50^{1011}-1011.50^{1010}+\ldots-1) \\\\& + (10^{1011}-1011.10^{1010}+\ldots . .-1) \\\\&= 5 m-1+5 n-1=5(m+n)-2 \\\\&= 5(m+n)-5+3=5(m+n-1)+3 \\\\&= 5 k+3 \\\\& \therefore \text { Remainder }=3\end{aligned}

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