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Home / Elementary Number Theory / 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

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

Proof: By division algorithm, there exist unique integers q′ and r′, such that

a = q′ b + r′,   0 ≤ r′ < b

⟹ a = q′ b + r′ + 2b – 2b

⟹ a = q′ b – 2b + r′ + 2b

⟹ a = (q′ – 2)b + r′ + 2b

Let, q = q′ – 2, r = r′ + 2b

∴ r, q unique integers.

Since 0 ≤ r′ < b, then

2b ≤ r′ + 2b < b + 2b

⟹ 2b ≤ r < 3b   (proved)

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