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In this paper we study Fleck's quotient $$F_p(n,r)=(-p)^{-\\lfloor(n-1)/(p-1)\\rfloor} \\sum_{k=r(mod p)}\\binom {n}{k}(-1)^k\\in Z.$$\n  We determine $F_p(n,r)$ mod $p$ completely by certain number-theoretic and combinatorial methods; consequently, if $2\\le n\\le p$ then $$\\sum_{k=1}^n(-1)^{pk-1}\\binom{pn-1}{pk-1} \\equiv(n-1)!B_{p-n}p^n (mod p^{n+1}),$$ where $B_0,B_1,...$ are Bernoulli numbers. 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