 So remember that if real numbers a and b are multiplicative inverses, then the product ab is equal to 1. We can extend this idea to congruences and define multiplicative inverses mod n. a and b are multiplicative inverses mod n when a times b is congruent to 1 mod n. Now finding a multiplicative inverse is a special case of solving a congruence. So for example let's try to find the inverse of 7 mod 26. So we need to find the multiplicative inverse of 7 mod 26 so we want to solve the congruence 7a congruent to 1 mod 26. So if 7a is congruent to 1 mod 26, then we know that 7a is 1 more than a multiple of 26. So we'll use our Diathontine method of solving this and get a equal to 3b plus 5b plus 1 divided by 7. So we want 5b plus 1 over 7 to be an integer and so we need some integer c to be equal to 5b plus 1 and solving this for b we find b is equal to c plus 2c minus 1 divided by 5. Again we want 2c minus 1 over 5 to be an integer and so we need some integer d to be equal to 2c minus 1 divided by 5. Applying our Diathontine method we get c is equal to 2d plus d plus 1 divided by 2. Again we want d plus 1 over 2 to be an integer so we need some integer e to be equal to d plus 1 over 2. We'll apply our Diathontine method and find d is equal to 2e minus 1 and so now we have our system and if e is an integer everything else will be as well. Now since our first computation is to find 2e minus 1 we'll make e equal to 1. So if e equals 1 then d is 1. If d is equal to 1 then c is 3. If c is equal to 3 then b is 4. If b is equal to 4 then a is 15 and so the multiplicative inverse of 7 mod 26 is equal to 15.