Suppose that n is a positive integer which is relatively prime to 10. Then for every positive integer a which is less than n and relatively prime to n, the decimal expansion of a is purely periodic. The length of the period is equal to the ________ of __________ in the group .

Two such fractions a and b have decimal expansions which are shifts of each other (that is, using the same digits) if and only if a and b lie in the same ________ of ________ in _________.

1/7 4/7 = 0.142857 = 0.571428 2/7 5/7 = 0.285714 = 0.714285 3/7 6/7 = 0.428571 = 0.857142 1/11 = 0.09 6/11 = 0.54 2/11 = 0.18 7/11 = 0.63 3/11 = 0.27 8/11 = 0.72 4/11 = 0.36 9/11 = 0.81 5/11 = 0.45 10/11 = 0.90 1/13 4/13 7/13 10/13 = 0.076923 = 0.307692 = 0.538461 = 0.769230 2/13 5/13 8/13 11/13 = 0.153846 = 0.384615 = 0.615384 = 0.846153 3/13 6/13 9/13 12/13 = 0.230769 0.461538 0.692307 = 0.923076 5/37 1/37 = 0.027 = 0.135 9/37 = 0.243 13/37 = 0.351 17/37 = 0.459 21/37 = 0.567 25/37 = 0.675 29/37 = 0.783 33/37 = 0.891 2/37 = 0.054 6/37 = 0.162 10/37 = 0.270 14/37 = 0.378 18/37 = 0.486 22/37 = 0.594 26/37 = 0.702 30/37 = 0.810 34/37 = 0.918 3/37 0.081 7/37 = 0.189 11/37 0.297 15/37 = 0.405 19/37 = 0.513 23/37 = 0.621 27/37 = 0.729 31/37 = 0.837 35/37 = 0.945 4/37 = 0.108 8/37 = 0.216 12/37 = 0.324 16/37 = 0.432 20/37 -0.540 24/37 = 0.648 28/37 = 0.756 32/37 = 0.864 36/37 = 0.972 1/41 5/41 9/41 13/41 17/41 21/41 25/41 29/41 33/41 37/41 = 0.02439 = 0.12195 = 0.21951 = 0.31707 = 0.41463 = 0.51219 = 0.60975 = 0.70731 = 0.80487 = 0.90243 2/41 = 0.04878 6/41 = 0.14634 10/41 = 0.24390 14/41 = 0.34146 18/41 = 0.43902 22/41 = 0.53658 26/41 = 0.63414 30/41 = 0.73170 34/41 = 0.82926 38/41 = 0.92682 3/41 = 0.07317 7/41 = 0.17073 11/41 = 0.26829 15/41 = 0.36585 19/41 = 0.46341 23/41 = 0.56097 27/41 = 0.65853 31/41 = 0.75609 35/41 = 0.85365 39/41 = 0.95121 4/41 8/41 12/41 16/41 20/41 24/41 28/41 32/41 36/41 40/41 = 0.09756 = 0.19512 = 0.29268 0.39024 = 0.48780 = 0.58536 0.68292 = 0.78048 = 0.87804 = 0.97560 Suppose that n is a positive integer which is relatively prime to 10. Then for every positive integer a which is less than n and relatively prime to n, the decimal expansion of a is purely periodic. The length of the period is equal to the of in the group Two such fractions a and b have decimal expansions which are shifts of each other (that is, using the same digits) if and only if a and b lie in the same of in 1/7 4/7 = 0.142857 = 0.571428 2/7 5/7 = 0.285714 = 0.714285 3/7 6/7 = 0.428571 = 0.857142 1/11 = 0.09 6/11 = 0.54 2/11 = 0.18 7/11 = 0.63 3/11 = 0.27 8/11 = 0.72 4/11 = 0.36 9/11 = 0.81 5/11 = 0.45 10/11 = 0.90 1/13 4/13 7/13 10/13 = 0.076923 = 0.307692 = 0.538461 = 0.769230 2/13 5/13 8/13 11/13 = 0.153846 = 0.384615 = 0.615384 = 0.846153 3/13 6/13 9/13 12/13 = 0.230769 0.461538 0.692307 = 0.923076 5/37 1/37 = 0.027 = 0.135 9/37 = 0.243 13/37 = 0.351 17/37 = 0.459 21/37 = 0.567 25/37 = 0.675 29/37 = 0.783 33/37 = 0.891 2/37 = 0.054 6/37 = 0.162 10/37 = 0.270 14/37 = 0.378 18/37 = 0.486 22/37 = 0.594 26/37 = 0.702 30/37 = 0.810 34/37 = 0.918 3/37 0.081 7/37 = 0.189 11/37 0.297 15/37 = 0.405 19/37 = 0.513 23/37 = 0.621 27/37 = 0.729 31/37 = 0.837 35/37 = 0.945 4/37 = 0.108 8/37 = 0.216 12/37 = 0.324 16/37 = 0.432 20/37 -0.540 24/37 = 0.648 28/37 = 0.756 32/37 = 0.864 36/37 = 0.972 1/41 5/41 9/41 13/41 17/41 21/41 25/41 29/41 33/41 37/41 = 0.02439 = 0.12195 = 0.21951 = 0.31707 = 0.41463 = 0.51219 = 0.60975 = 0.70731 = 0.80487 = 0.90243 2/41 = 0.04878 6/41 = 0.14634 10/41 = 0.24390 14/41 = 0.34146 18/41 = 0.43902 22/41 = 0.53658 26/41 = 0.63414 30/41 = 0.73170 34/41 = 0.82926 38/41 = 0.92682 3/41 = 0.07317 7/41 = 0.17073 11/41 = 0.26829 15/41 = 0.36585 19/41 = 0.46341 23/41 = 0.56097 27/41 = 0.65853 31/41 = 0.75609 35/41 = 0.85365 39/41 = 0.95121 4/41 8/41 12/41 16/41 20/41 24/41 28/41 32/41 36/41 40/41 = 0.09756 = 0.19512 = 0.29268 0.39024 = 0.48780 = 0.58536 0.68292 = 0.78048 = 0.87804 = 0.97560 Suppose that n is a positive integer which is relatively prime to 10. Then for every positive integer a which is less than n and relatively prime to n, the decimal expansion of a is purely periodic. The length of the period is equal to the of in the group Two such fractions a and b have decimal expansions which are shifts of each other (that is, using the same digits) if and only if a and b lie in the same of in