Question

Let 𝛂 be the remainder (22)\(^{2022}\) + (2022)\(^{22}\) is divided by 3 and ꞵ be the remainder when the same is divided by 7 then 𝛂\(^2\) + ꞵ\(^2\) is?

Answer 1

Correct Answer: 5

To solve this problem, we need to find 𝛂 and ꞵ as described and then compute 𝛂²+ꞵ². First, determine the remainder of (22)²⁰²²+2022²² when divided by 3. By Fermat's Little Theorem, if p is a prime, then a^p-1≡1 (mod p) for any integer a not divisible by p. Applying this mod 3:
(22)≡1 (mod 3). Thus, (22)²⁰²²≡1²⁰²²≡1 (mod 3).
For 2022²², note that 2022≡0 (mod 3), so 2022²²≡0 (mod 3).
Therefore, (22)²⁰²²+2022²²≡1+0≡1 (mod 3) and 𝛂=1.
Next, find the remainder when dividing by 7. Again using Fermat's Little Theorem: (22)≡1(mod 7), so (22)²⁰²²≡1 (mod 7).
To find 2022²² mod 7, use 2022≡6(mod 7). Notice 6≡-1(mod 7), so 6²²≡(-1)²²≡1(mod 7).
Therefore, (22)²⁰²²+2022²²≡1+1≡2(mod 7) and ꞵ=2.
Now, calculate 𝛂²+ꞵ²:
𝛂²=1²=1 and ꞵ²=2²=4,
Resulting in 𝛂²+ꞵ²=1+4=5.
The solution, 5, clearly falls within the range 5 to 5 as expected.