Determine all possible positive integer(s) N < 10,000, such that:

2^N = 88 (mod 167), and:

2^N = 70 (mod 83)

**** For an extra challenge, solve this problem without using a computer program.

For positive integer N, such that N < 10000 and
2^N = 88 (mod 167),
N = 12+83k : 0 <= k < INT(10000/83)+1

{12, 95, 178, 261, 344, 427, 510, 593, 676, 759, 842, 925, 1008, 1091, 1174, 1257, 1340, 1423, 1506, 1589, 1672, 1755, 1838, 1921, 2004, 2087, 2170, 2253, 2336, 2419, 2502, 2585, 2668, 2751, 2834, 2917, 3000, 3083, 3166, 3249, 3332, 3415, 3498, 3581, 3664, 3747, 3830, 3913, 3996, 4079, 4162, 4245, 4328, 4411, 4494, 4577, 4660, 4743, 4826, 4909, 4992, 5075, 5158, 5241, 5324, 5407, 5490, 5573, 5656, 5739, 5822, 5905, 5988, 6071, 6154, 6237, 6320, 6403, 6486, 6569, 6652, 6735, 6818, 6901, 6984, 7067, 7150, 7233, 7316, 7399, 7482, 7565, 7648, 7731, 7814, 7897, 7980, 8063, 8146, 8229, 8312, 8395, 8478, 8561, 8644, 8727, 8810, 8893, 8976, 9059, 9142, 9225, 9308, 9391, 9474, 9557, 9640, 9723, 9806, 9889, 9972}


For positive integer N, such that N < 10000 and
2^N = 70 (mod 83),
N = 36+82k : 0 <= k < INT(10000/82)+1

{36, 118, 200, 282, 364, 446, 528, 610, 692, 774, 856, 938, 1020, 1102, 1184, 1266, 1348, 1430, 1512, 1594, 1676, 1758, 1840, 1922, 2004, 2086, 2168, 2250, 2332, 2414, 2496, 2578, 2660, 2742, 2824, 2906, 2988, 3070, 3152, 3234, 3316, 3398, 3480, 3562, 3644, 3726, 3808, 3890, 3972, 4054, 4136, 4218, 4300, 4382, 4464, 4546, 4628, 4710, 4792, 4874, 4956, 5038, 5120, 5202, 5284, 5366, 5448, 5530, 5612, 5694, 5776, 5858, 5940, 6022, 6104, 6186, 6268, 6350, 6432, 6514, 6596, 6678, 6760, 6842, 6924, 7006, 7088, 7170, 7252, 7334, 7416, 7498, 7580, 7662, 7744, 7826, 7908, 7990, 8072, 8154, 8236, 8318, 8400, 8482, 8564, 8646, 8728, 8810, 8892, 8974, 9056, 9138, 9220, 9302, 9384, 9466, 9548, 9630, 9712, 9794, 9876, 9958}

Only 2004 and 8810 are common to both sets, therefore they are the two solutions.

Posted by Dej Mar on 2011-04-25 17:24:49