3 from that of line 1. This completes the first round. Consider next the ratio 7/4, whose integer part is [7/4] = 1. In step 5, we multiply the results of line 4 by 1, and in step 6, we subtract the result of line 5 from that of line 2. This completes the second round. Consider next the ratio 4/3, whose integer part is Modular arithmetic and Euler's theorem Table T1 Extended Euclidian algorithm to find 7-1 mod 39. 7 = 35 X 1 = 5 39 - 35 = 4 39 - 5 = 34 1 x 4 = 4 1 X 34 = 34 7 4 = 3 1 - 34 = -33 1 x 3 = 3 1 X (-33) = -33 4 3 = 1 34 - (-33) = 67 = 28 Table T2 Extended Euclidian algorithm to find 5-1 mod 16 272. 16 272 16272 3254 X 5 = 16270 3254 X 1 = 3 254 16272 - 16 270 = 2 16272 - 3254 = 13018 2 X 2 = 4 2 X 13018 = 26036 5 - 4 = 1 1 - 26036 = -26035 [4/3] = 1. In step 7, we multiply the results of line 6 by 1, and in step 8, we subtract the result of line 7 from that of line 4. If we have obtained 1 in the first column (line 8), we have finished. When positive, the number to the right is the inverse we have been looking for. Here, we find that the inverse of 7 is 28. Proof: 7 X 28 = 196 = 1 (modulo Consider next the example in Table T2, which concerns the search of 5-1 mod 16272. In steps 1 and 2, we fill out the table as shown. We first divide 8064 by 7, to find that the integer part of the result is [16272/5] = 3254. In step 3, we multiply line 2 by 1612, and in step 4 we subtract the result of line 3 from that of line 1. This completes the first round. Consider next the ratio 5/2, whose integer part is [5/2] = 2. In step 5, we multiply the results of line 4 by 2, and in step 6, we subtract the result of line 5 from that of line 2. This completes the search, since we have obtained 1 in the first column (line 6). However, the number in the right column, b = - 26035 is negative. In this case, the inverse is the complement km + b, namely 5-1 = 2 x 16272 - 26035 = 6509. The proof is that 5 X 6509 = 2545 = (2 X 16 272) + 1 = 1 (modulo 16 272). This numerical value will be used in the main text concerning my numeric