# Another fact on factorial

Number Theory Level 4

Let $$T(x)$$ denote the number of trailing zeros in $$\prod_{n=1}^{x} n!$$ for a positive integer $$x$$.

Let $$\{x_1,x_2,\cdots,x_p\}$$ be the set of all positive integers $$x_i$$ which satisfy the equation

$T(x_i)-x_i+1=0$

Find $T\left(\sum_{i=1}^p x_i\right)-\sum_{i=1}^p x_i+1$

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