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A positive integer is said to be special if the sum of the remainders obtained when it is divided by five consecutive positive integers is $$32$$. For example, $$24$$ is special since when divided by $$11,12,13,14,15$$ the remainders are $$2,0,11,10,9$$ and these add up to $$32$$. The smallest positive integer that is special is $$\underline{\qquad}$$.
Correct Answer: 8
For every integer below $$7$$, the sum of five remainders is at most $$30$$. For $$7$$, the possible largest patterns are $$0,7,7,7,7$$ or $$7,7,7,7,7$$, whose sums are $$28$$ and $$35$$, not $$32$$. For $$8$$, division by $$8,9,10,11,12$$ gives remainders $$0,8,8,8,8$$, whose sum is $$32$$.
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