Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem K. 107. (December 2006)

K. 107. If two digits of the four-digit number 1234 are cancelled in every possible way and the remaining digits are read as a two-digit number, the numbers 12, 13, 14, 23, 24, 34 are obtained. Their sum is 120. Find a four-digit number for which this sum is a) 540; b) 220.

(6 pont)

Deadline expired on January 10, 2007.


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

Megoldás. a) pl. 9876. b) Nincs ilyen. Legyen a keresett négyjegyű szám \overline{abcd} alakú, ekkor a keresett összeg \overline{ab}+\overline{ac}+\overline{ad}+\overline{bc}+
\overline{bd}+\overline{cd}=10a+b+10a+c+10a+d+10b+c+10b+d+10c+d=30a+21b+12c+3d. Ez az összeg osztható 3-mal, a 220 pedig nem. Tehát az ilyen módon előállított számok összege nem lehet 220.


Statistics:

149 students sent a solution.
6 points:95 students.
5 points:7 students.
4 points:12 students.
3 points:33 students.
1 point:1 student.
Unfair, not evaluated:1 solution.

Problems in Mathematics of KöMaL, December 2006