2021 JMPSC Accuracy Problems/Problem 15

Problem

For all positive integers $n,$ define the function $f(n)$ to output $4\underbrace{777 \cdots 7}_{n\ \text{sevens}}5.$ For example, $f(1)=475$ , $f(2)=4775$ , and $f(3)=47775.$ Find the last three digits of $\frac{f(1)+f(2)+ \cdots + f(100)}{25}.$

Solution

Notice that $\frac{f(1)}{25} = 19$ , $\frac{f(2)}{25} = 191$ , $\frac{f(3)}{25} = 1911$ , and $\frac{f(n)}{25}$ ends in $111$ for all $n \ge 4$ . So, the last 3 digits of $\frac{f(1)+f(2)+ \cdots + f(100)}{25}$ are the last $3$ digits of $19 + 191 + 911 + 111 \cdot 97$ , which are $\boxed{888}$ .

~Mathdreams

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2021 JMPSC Accuracy Problems/Problem 15

Problem

Solution