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Never prime! (3) (Posted on 2011-02-25) Difficulty: 3 of 5
The value of the smallest positive base ten integer that cannot be changed into a prime by changing a single digit was determined in Never prime!.

M denotes the smallest value of a base N positive integer that cannot be changed into a prime by changing a single digit.

For the values of N drawn at random between 51 and 100 inclusively, determine the probability that the first digit of M (reading left to right) is 3.

Note: N is a positive integer and, M cannot contain any leading zero, and the first digit of M (reading left to right) cannot be changed to a zero.

No Solution Yet Submitted by K Sengupta    
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solution Comment 1 of 1
Continuing from the work done on Never Prime! (2), the following numbers are, to the best of my knowledge, the smallest desired values for base 51 to 100.

51    31416 {12, 4, 0}
52    31408 {11, 32, 0}
53   107378 {38, 12, 0}
54    89694 {30, 41, 0}
55    31405 {10, 21, 0}
56   102704 {32, 42, 0}
57    31407 {9, 38, 0}
58   173362 {51, 31, 0}
59   107380 {30, 50, 0}
60   134520 {37, 22, 0}
61   173362 {46, 36, 0}
62   155930 {40, 35, 0}
63   155925 {39, 18, 0}
64   188032 {45, 58, 0}
65   155935 {36, 59, 0}
66   188034 {43, 11, 0}
67   332320 {1, 7, 2, 0}
68   155924 {33, 49, 0}
69   360663 {1, 6, 52, 0}
70   370300 {1, 5, 40, 0}
71   370265 {1, 2, 32, 0}
72   338040 {65, 15, 0}
73   155928 {29, 19, 0}
74   370296 {67, 46, 0}
75   155925 {27, 54, 0}
76   370272 {64, 8, 0}
77   155925 {26, 23, 0}
78   155922 {25, 49, 0}
79   370273 {59, 26, 0}
80   576800 {1, 10, 10, 0}
81   155925 {23, 62, 0}
82  1053864 {1, 74, 60, 0}
83   370263 {53, 62, 0}
84   370272 {52, 40, 0}
85   360655 {49, 78, 0}
86  1313478 {2, 5, 51, 0}
87   370272 {48, 80, 0}
88   927872 {1, 31, 72, 0}
89  1140001 {1, 54, 82, 0}
90   492120 {60, 68, 0}
91   370279 {44, 65, 0}
92  1098848 {1, 37, 76, 0}
93   360654 {41, 65, 0}
94   370266 {41, 85, 0}
95  1313470 {1, 50, 51, 0}
96   370272 {40, 17, 0}
97  1357224 {1, 47, 24, 0}
98  1357202 {1, 43, 31, 0}
99  1444311 {1, 48, 36, 0}
100 1671800 {1, 67, 18, 0}

Of those 50 numbers, 20 have a first digit of 3. That gives a probability of 0.4 or 40%.

  Posted by Justin on 2011-03-02 09:58:41
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