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

Solution 2 (More Algebraic)

$\sum_{n=1}^{100} f(n) = 5(100)+70(\underbrace{1+11+111+1111+ \cdots}_{\text{100 1s}}) + 100(44444 \cdots )$ We only care about the last $3$ digits, so we evaluate $1+11+111+1111+ \cdots$ . Note the expression is simply $1(100)+10(99)+100(98)+1000(97)+ \cdots + 10^{100}$ , so factoring a $10$ we have $1(10)+99+10(98)+ \cdots + 10^{99}$ . Now, we can divide by $25$ to get $20+28(1(10)+99+10(98)+100(97) \cdots)+4(444444 \cdots )$ Evaluate the last $3$ digits to get $20+28(10+99+980+700)+4(444)=\boxed{888} \mod 1000$ $\linebreak$ ~Geometry285

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2021 JMPSC Accuracy Problems/Problem 15

Problem

Solution

Solution 2 (More Algebraic)