Problem K. 603. (November 2018)

K. 603. I have a two-digit number in mind. Let \(\displaystyle S\) denote the sum of the digits, and let \(\displaystyle P\) denote their product. What may be my number if it is equal to \(\displaystyle P+S\)?

(6 pont)

Deadline expired on December 10, 2018.

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

Megoldás. Legyen a kétjegyű \(\displaystyle n\) szám első számjegye \(\displaystyle a\), második \(\displaystyle b\), ekkor \(\displaystyle P = ab\), \(\displaystyle S = a+b\), \(\displaystyle n = 10a + b\), azaz a \(\displaystyle 10a + b = ab +a + b\) egyenlet megoldását keressünk. Ebből \(\displaystyle ab = 9a\) és mivel \(\displaystyle a\) nem lehet \(\displaystyle 0\), így \(\displaystyle b = 9\). Tehát a \(\displaystyle 19\), \(\displaystyle 29\), \(\displaystyle 39\), \(\displaystyle 49\), \(\displaystyle 59\), \(\displaystyle 69\), \(\displaystyle 79\), \(\displaystyle 89\) és \(\displaystyle 99\) számokra gondolhattam.

Statistics:

220 students sent a solution.
6 points:	73 students.
5 points:	66 students.
4 points:	5 students.
3 points:	10 students.
2 points:	22 students.
1 point:	7 students.
0 point:	4 students.
Not shown because of missing birth date or parental permission:	33 solutions.

Problems in Mathematics of KöMaL, November 2018