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Rooting for two digits (Posted on 2011-06-24) Difficulty: 3 of 5
Determine the probability that for a positive integer N chosen at random between 1000 (base ten) and 9999 (base ten) inclusively, the number formed by the last two digits (reading left to right) of N is a divisor of [√N].

Notes:

(I) [x] denotes the greatest integer ≤ x

(II) Reading from right to left, the second digit of N is nonzero.

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

DEFDBL A-Z
FOR n = 1000 TO 9999
  last2 = n MOD 100
  ct = ct + 1
  sr = INT(SQR(n))
  IF last2 > 9 THEN ct2 = ct2 + 1
  IF last2 > 0 THEN
  IF sr MOD last2 = 0 THEN
    hit = hit + 1
    IF last2 > 9 THEN
      hit2 = hit2 + 1
      PRINT n, sr
      IF hit2 MOD 45 = 0 THEN DO: LOOP UNTIL INKEY$ > "": PRINT
    END IF
  END IF
  END IF
NEXT
PRINT hit; ct, hit2; ct2

finds that 236 of the 8100 numbers eligible (the 9000 in the range minus the 900 which have a zero in the tens position) meet the divisibility criterion.  If not for the non-zero requirement for the tens position, 503 out of 9000 would meet the divisibility requirement.  Thus the answer is 236/8100 = 59/2025.


   N        [sqrt(N)]
 1032          32
 1111          33
 1133          33
 1217          34
 1235          35
 1312          36
 1318          36
 1336          36
 1437          37
 1519          38
 1539          39
 1610          40
 1620          40
 1640          40
 1741          41
 1814          42
 1821          42
 1842          42
 1944          44
 2011          44
 2022          44
 2045          45
 2115          45
 2123          46
 2146          46
 2247          47
 2312          48
 2316          48
 2324          48
 2348          48
 2449          49
 2510          50
 2525          50
 2550          50
 2617          51
 2651          51
 2713          52
 2726          52
 2752          52
 2853          53
 2918          54
 2927          54
 2954          54
 3018          54
 3055          55
 3111          55
 3156          56
 3214          56
 3228          56
 3257          57
 3319          57
 3357          57
 3429          58
 3458          58
 3559          59
 3610          60
 3612          60
 3615          60
 3620          60
 3630          60
 3660          60
 3710          60
 3712          60
 3715          60
 3720          60
 3761          61
 3862          62
 3931          62
 3962          62
 4021          63
 4063          63
 4116          64
 4132          64
 4164          64
 4216          64
 4265          65
 4313          65
 4366          66
 4411          66
 4422          66
 4433          66
 4466          66
 4567          67
 4634          68
 4668          68
 4717          68
 4734          68
 4769          69
 4823          69
 4869          69
 4910          70
 4914          70
 4935          70
 4970          70
 5010          70
 5014          70
 5035          70
 5071          71
 5171          71
 5212          72
 5218          72
 5224          72
 5236          72
 5272          72
 5312          72
 5318          72
 5324          72
 5373          73
 5473          73
 5537          74
 5574          74
 5625          75
 5675          75
 5715          75
 5725          75
 5775          75
 5776          76
 5819          76
 5838          76
 5876          76
 5919          76
 5977          77
 6011          77
 6077          77
 6113          78
 6126          78
 6139          78
 6178          78
 6213          78
 6226          78
 6239          78
 6279          79
 6379          79
 6410          80
 6416          80
 6420          80
 6440          80
 6480          80
 6510          80
 6516          80
 6520          80
 6540          80
 6581          81
 6627          81
 6681          81
 6741          82
 6782          82
 6841          82
 6882          82
 6983          83
 7084          84
 7112          84
 7114          84
 7121          84
 7128          84
 7142          84
 7184          84
 7212          84
 7214          84
 7221          84
 7285          85
 7317          85
 7385          85
 7443          86
 7486          86
 7543          86
 7587          87
 7629          87
 7687          87
 7729          87
 7744          88
 7788          88
 7811          88
 7822          88
 7844          88
 7888          88
 7911          88
 7989          89
 8089          89
 8110          90
 8115          90
 8118          90
 8130          90
 8145          90
 8190          90
 8210          90
 8215          90
 8218          90
 8230          90
 8245          90
 8291          91
 8313          91
 8391          91
 8413          91
 8492          92
 8523          92
 8546          92
 8592          92
 8623          92
 8646          92
 8693          93
 8731          93
 8793          93
 8831          93
 8847          94
 8894          94
 8947          94
 8994          94
 9095          95
 9119          95
 9195          95
 9216          96
 9224          96
 9232          96
 9248          96
 9296          96
 9312          96
 9316          96
 9324          96
 9332          96
 9348          96
 9396          96
 9497          97
 9597          97
 9614          98
 9649          98
 9698          98
 9714          98
 9749          98
 9798          98
 9811          99
 9833          99
 9899          99
 9911          99
 9933          99
 9999          99

  Posted by Charlie on 2011-06-24 17:20:33
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