Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
Already signed up?
New to KöMaL?

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