Mathematical and Physical Journal
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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.


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Problems in Mathematics of KöMaL, September 2018