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.
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
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Posted by Charlie
on 2011-06-24 17:20:33 |
computer solution
