Question

Let the number  \((22)^{2022}\) + \((2022)^{22}\)  leave the remainder \( \alpha \) when divided by 3 and \( \beta \) when divided by 7. Then \( (\alpha^2 + \beta^2) \) is equal to:}

• A. 5
• B. 20
• C. 10
• D. 13

Answer 1

The Correct Option is A

Step 1: Find \( \alpha \) modulo 3 We need to find the remainder when \( (22)^{2022} + (2022)^{22} \) is divided by 3. We know: \[ (21 + 1)^{2022} + (2022)^{22} \equiv 3k + 1 \quad \Rightarrow \quad \alpha = 1 \] Step 2: Find \( \beta \) modulo 7 Now we find the remainder when \( (22)^{2022} + (2022)^{22} \) is divided by 7. We get: \[ (21 + 1)^{2022} + (2023 - 1)^{22} \equiv 7k + 1 \quad \Rightarrow \quad \beta = 2 \] Step 3: Compute \( \alpha^2 + \beta^2 \) Now we compute: \[ \alpha^2 + \beta^2 = 1^2 + 2^2 = 1 + 4 = 5 \] Thus, the value of \( \alpha^2 + \beta^2 \) is \( 5 \).