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We determine the explicit formulas for the sum of products of homogeneous multiple
harmonic sums $\sum_{k=1}^n \prod_{j=1}^r H_k(\{1\}^{\lambda_j})$ when 
$\sum_{j=1}^r \lambda_j\leq 5$. 
We apply these identities to the study of two congruences modulo a power of a prime.


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