A positive integer is called special if the sum of the remainders obtained when it is divided by five consecutive positive integers is $$32$$. The smallest positive integer that is special is

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 $$32$$.