JAMIA 2021 — Mathematics PYQ

Step 1: Understand the given condition

The problem states that $x$ divided by $4$ leaves a remainder of $3$. This can be written as:

This means $x$ can be expressed in the form:

Step 2: Simplify the base expression modulo 8

We need to find the remainder of $(2020+x{)}^{2022}$ when divided by $8$. Let's first look at the terms inside the parentheses:

1. For $2020$:

   $$2020=252\times 8+4$$

   So, $2020\equiv 4(\mathrm{mod}8)$.

2. For $x$:

   Since $x=4k+3$, we have two cases based on whether $k$ is even or odd:

   • If $k$ is even ($k=2m$): $x=4(2m)+3=8m+3\equiv 3(\mathrm{mod}8)$.

   • If $k$ is odd ($k=2m+1$): $x=4(2m+1)+3=8m+4+3=8m+7\equiv 7(\mathrm{mod}8)$.

Step 3: Analyze $(2020+x)(\mathrm{mod}8)$

Now, let's combine these:

• Case 1 (If $k$ is even):

  $$2020+x\equiv 4+3=7(\mathrm{mod}8)$$

• Case 2 (If $k$ is odd):

  $$2020+x\equiv 4+7=11\equiv 3(\mathrm{mod}8)$$

Step 4: Calculate the power modulo 8

We need to find $(2020+x{)}^{2022}(\mathrm{mod}8)$. Note that the exponent $2022$ is an even number.

• In Case 1 ($7(\mathrm{mod}8)$):

  $$7^2=49\equiv 1(\mathrm{mod}8)$$

  Therefore, $7^{2022}=(7^2{)}^{1011}\equiv 1^{1011}\equiv 1(\mathrm{mod}8)$.

• In Case 2 ($3(\mathrm{mod}8)$):

  $$3^2=9\equiv 1(\mathrm{mod}8)$$

  Therefore, $3^{2022}=(3^2{)}^{1011}\equiv 1^{1011}\equiv 1(\mathrm{mod}8)$.

In both possible cases for $x$, the result is the same.

Final Answer:

The remainder when $(2020+x{)}^{2022}$ is divided by $8$ is $1$.