1. It turns out that 247248 mod 249 = 4. From this, you can conclude at least one of following: 247 is not prime; 247 is prime; 249 is not prime; or 249 is prime. Which one(s)? Explain 2. Reprove the...

1. It turns out that 2472⁴⁸
mod 249 = 4. From this, you can conclude at least one of following: 247 is not prime; 247 is prime; 249 is not prime; or 249 is prime. Which one(s)? Explain

2. Reprove the general version of the Chinese Remainder Theorem with single constructive argument, as in the 2-congruence case, instead of using induction. Namely, assume n1, n2, . . . , nk are pairwise relatively prime, and let ai ∈ Zni . Let N := ∏ k i=1 ni . Let Ni := N/ni (more precisely, let Ni be the product of all njs except ni) and let di be the multiplicative inverse of Ni in Zni . Prove that x := ∑ k i=1 aiNidi satisfies the congruence x mod ni = ai for all 1 ≤ i ≤ k.