Each of p and q is a
positive integer with gcd(p, q)=1.
Determine the minimum value of p+q such that:
- The first four digits immediately following the decimal point in the base ten expansion of p/q + q/p is 2013.
The answer is 169 as found below:
DECLARE FUNCTION gcd# (a#, b#)
DEFDBL A-Z
CLS
FOR t = 3 TO 99999
FOR p = 1 TO t / 2
q = t - p
v$ = STR$(p / q + q / p)
ix = INSTR(v$, ".")
IF ix > 0 AND gcd(p, q) = 1 THEN
IF MID$(v$, ix + 1, 4) = "2013" THEN
PRINT USING "#### #### ####.########## #####"; p; q; VAL(v$); t
ct = ct + 1
END IF
END IF
NEXT
IF ct > 40 THEN END
NEXT
FUNCTION gcd (a, b)
dnd = a: dvr = b
DO
q = INT(dnd / dvr): r = dnd - q * dvr
dnd = dvr: dvr = r
LOOP UNTIL dvr = 0
gcd = dnd
END FUNCTION
finds the 41 lowest p+q satisfying the criterion:
p
and p/q + q/p p+q
q
66 103 2.2013827596 169
28 169 6.2013947591 197
87 248 3.2013811643 335
77 304 4.2013414217 381
157 245 2.2013258807 402
67 473 7.2013505412 540
223 348 2.2013427143 571
11 563 51.2013563701 574
25 604 24.2013907285 629
248 387 2.2013107444 635
17 683 40.2013607786 700
289 451 2.2013518594 740
32 709 22.2013839915 741
194 553 3.2013292071 747
339 529 2.2013037344 868
355 554 2.2013576041 909
380 593 2.2013357593 973
405 632 2.2013166120 1037
212 837 4.2013987512 1049
116 937 8.2013855666 1053
421 657 2.2013615477 1078
281 801 3.2013452935 1082
301 858 3.2013141897 1159
16 1155 72.2013528139 1171
129 1042 8.2013197637 1171
487 760 2.2013644223 1247
512 799 2.2013478763 1311
101 1224 12.2013282211 1325
537 838 2.2013328711 1375
127 1283 10.2013489545 1410
553 863 2.2013666109 1416
368 1049 3.2013537738 1417
289 1141 4.2013834765 1430
562 877 2.2013192013 1439
569 888 2.2013984547 1457
587 916 2.2013066960 1503
603 941 2.2013383314 1544
408 1163 3.2013070490 1571
619 966 2.2013683327 1585
327 1291 4.2013042541 1618
635 991 2.2013968234 1626
Edited on February 22, 2013, 2:07 pm
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Posted by Charlie
on 2013-02-22 14:06:08 |
computer solution
