Question: 3. Prove that there exist 135 consecutive integers, a1, a2, . . . ,a135, such that for every integer i, 1 S i S 135,

3. Prove that there exist 135 consecutive integers, a1, a2, . . . ,a135, such that for every integer i, 1 S i S 135, there is a perfect cube ki > 1 such that lei | ai. (Hint: Construct a system of simultaneous linear congruences and use the Generalized Chinese Remainder Theorem.)

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