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Point to 2013 (2) (Posted on 2013-02-22) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution | Comment 1 of 2

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
  Posted by Charlie on 2013-02-22 14:06:08

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