Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# 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