Trapped

My solution to this Problem is A salute to one of my 4 Idols $First\quad we\quad suppose\quad that\quad The\quad person\quad on\quad 19th\quad floor\quad presses\quad +8\quad Button\quad x\quad times\\ and\quad -13\quad button\quad y\quad times.\\ Every\quad time\quad he\quad presses\quad a\quad button\quad any\quad of\quad the\quad lift\quad shows\quad an\quad increment\quad of\quad 8\quad or\quad decrement\quad of\quad 13\\ This\quad procedure\quad reduces\quad the\quad sum\quad of\quad the\quad floors\quad lift\quad were\quad earlier\quad on\quad by\quad 13\quad or\quad increases\quad them\quad by\quad 8\\ First\quad the\quad sum\quad of\quad floors\quad lift\quad were\quad on\quad is-\\ 17+26+19+31=113\quad And\quad later\quad becomes\quad 21+22+23+24+25=115.So\quad there\quad is\quad a\quad net\quad increase\quad of\quad 2.This\quad gives\quad us\quad the\quad following\\ equation-\\ 8x-13y=2\quad -->8x-16y+3y=2\\ Dividing\quad both\quad sided\quad by\quad 8\quad we\quad have\quad 3y\quad must\quad leave\quad remainder\quad 2\quad when\quad divided\quad by\quad 8.\\ Also\quad note\quad that\quad x>y\\ So\quad y\quad must\quad be\quad in\quad form\quad of\quad 8k+6\quad so\quad y\quad can\quad be=6,14,22....\\ From\quad the\quad options\quad we\quad have\quad only\quad one\quad possible\quad value\quad i.e\quad 6.Putting\quad 6\quad we\quad have\quad \\ 8x-78=2--->8x=80--->x=10\quad so\quad x+y=16\quad which\quad is\quad our\quad answer.$

17+26+20+19+13=113. 21+22+...+25=115.

Every time 2 elevators go up the sum increases by 16. Every time 2 elevators go down the sum decreases by 26. So 16a-26b=115-113=2

8a-13b=1 The smallest solution is (5,3). So we need at least 2*(5+3)=16 moves.

Here is one possible way to do it in 16.

(17,19,20,26,31)

(25,27,20,26,31)

(25,14,20,13,31)

(25,14,20,0,18)

(25,14,20,8,26)

(25,22,20,16,26)

(25,22,28,16,34)

(25,22,36,24,34)

(25,22,23,24,21)