2021 WSMO Accuracy Round Problems/Problem 7

Problem

Find the value of $\sum_{n=1}^{100}\left(\sum_{i=1}^{n}r_i\right),$ where $r_i$ is the remainder when $2^i+3^i$ is divided by 10.

Solution 1

From Fermat's Little Theorem, we find that $a^n\pmod{10}$ is periodic in cycles of 4. This means that $a^n\equiv a^{n+4}\pmod{10}$ for all $a,n.$ Now, we will compute the first 4 values of $2^n+3^n\pmod{10}.$ $\begin{align*} 2^1+3^1\equiv2+3\equiv&\text{ }5\pmod{10}\\ 2^2+3^2\equiv4+9\equiv13\equiv&\text{ }3\pmod{10}\\ 2^3+3^3\equiv8+27\equiv35\equiv&\text{ }5\pmod{10}\\ 2^4+3^4\equiv16+81\equiv97\equiv&\text{ }7\pmod{10}\\ \end{align*}$ Thus,

\begin{align*}
	2^n+3^n\equiv&\text{ }5\pmod{10}\tag{for n\equiv1,3\pmod{4}}\\
  	2^n+3^n\equiv&\text{ }3\pmod{10}\tag{for n\equiv2\pmod{4}}\\
	2^n+3^n\equiv&\text{ }7\pmod{10}\tag{for n\equiv0\pmod{4}}\\  
\end{align*} (Error compiling LaTeX. Unknown error_msg)

This means that $\begin{align*} \sum_{n=1}^{100}\left(\sum_{i=1}^{n}r_i\right)&=\sum_{n=4a,1\leq a\leq25}(20a)+\sum_{n=4a+1,0\leq a\leq24}(20a+5)+\sum_{n=4a+2,0\leq a\leq24}(20a+8)+\sum_{n=4a+3,0\leq a\leq24}(20a+13)\\ &=\frac{25\cdot26}{2}\cdot20+\left(\frac{24\cdot25}{2}\cdot20+25\cdot5\right)+\left(\frac{24\cdot25}{2}\cdot20+25\cdot8\right)+\left(\frac{24\cdot25}{2}\cdot20+25\cdot13\right)\\ &=325\cdot20+300\cdot20\cdot3+25\cdot(5+8+13)\\ &=6500+6000\cdot3+25\cdot26\\ &=6500+18000+650\\ &=\boxed{25150}\\ \end{align*}$ ~pinkpig

Solution 2

We will begin by examining $2^i\text{ (mod }10\text{)}$ and $3^i\text{ (mod }10\text{)}$ :

$2^1 \equiv 2\text{ (mod }10\text{)}$ $2^2 \equiv 4\text{ (mod }10\text{)}$ $2^3 \equiv 8\text{ (mod }10\text{)}$ $2^4 \equiv 6\text{ (mod }10\text{)}$ $...$ and $3^1 \equiv 3\text{ (mod }10\text{)}$ $3^2 \equiv 9\text{ (mod }10\text{)}$ $3^3 \equiv 7\text{ (mod }10\text{)}$ $3^4 \equiv 1\text{ (mod }10\text{)}$ $...$

From this, we can note that:

$r_i = 5 \text{, } i \equiv 1 \text{ (mod }4\text{)}$ $r_i = 3 \text{, } i \equiv 2 \text{ (mod }4\text{)}$ $r_i = 5 \text{, } i \equiv 3 \text{ (mod }4\text{)}$ $r_i = 7 \text{, } i \equiv 0 \text{ (mod }4\text{)}$

We can simplify our sum as follows:

$\sum_{n=1}^{100}\left(\sum_{i=1}^{n}r_i\right)$ $=100r_1 + 99r_2 + 98r_3 + ... + r_{100}$ $=(100+96+...+4)r_1 + (99+95+...+3)r_2 + (98+94+...+2)r_3 + (97+93+...+1)r_4$

Note that $100+96+...+4 = 4(25+24+...+1) = 4(\frac{25 \cdot 26}{2}) = 1300$ :

$=1300r_1 + (1300-25)r_2 + (1300-50)r_3 + (1300-75)r_4$ $=5(1300) + 3(1275) + 5(1250) + 7(1225)$ $=\boxed{25150}$

~BigKahuna227

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2021 WSMO Accuracy Round Problems/Problem 7

Problem

Solution 1

Solution 2