1) i) Prove that if a and b are integers, with b>0, then there exist unique integers q and r satisfying a = qb+r, where 2b ≤r < 3b. ii) Verify that 32 divides (a²+3)(a² + 7) if a is an odd integer. iii) Find ged(112, 2021) and show that ged(112, 2021) = 112r+2021y for suitable integers x, y by using the Euclidean Algorithm.

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Q: Show that if x is an integer, then gcd(x + 1, x2 – x + 1) divides 3. -

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A: Step 1: Step 2: Step 3: Step 4:

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