Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem C. 1490. (September 2018)

C. 1490. What is the remainder if the number $\displaystyle N=863\underbrace{99\ldots 9}_\text{2018 times}$ (with 2018 digits of 9 at the end) is divided by 32?

(5 pont)

Deadline expired on October 10, 2018.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. $\displaystyle N+1=864\underbrace{000…0}_\text{2018 db}=864\cdot10^{2018}=2^5\cdot3^3\cdot10^{2018}$. A $\displaystyle k=3^3\cdot10^{2018}$ jelöléssel $\displaystyle N+1=2^5\cdot k=32k$, amiből

$\displaystyle N=32k-1=32k-32+31=32(k-1)+31.$

Tehát az $\displaystyle N$ szám $\displaystyle 31$ maradékot ad $\displaystyle 32$-vel osztva.

### Statistics:

 328 students sent a solution. 5 points: 201 students. 4 points: 35 students. 3 points: 60 students. 2 points: 6 students. 1 point: 2 students. 0 point: 8 students. Unfair, not evaluated: 1 solutions. Not shown because of missing birth date or parental permission: 15 solutions.

Problems in Mathematics of KöMaL, September 2018