Answer :
Final answer:
50 is the inverse of 5 modulo 83 because their product gives a remainder of 1 when divided by 83, as does the product of 12 and 7, confirming 12 as the inverse of 7 modulo 83. The inverse of 35 modulo 83 is found to be 71, as it satisfies the equation 35x ≡ 1 (mod 83).
Explanation:
To demonstrate that 50 is the inverse of 5 modulo 83, and 12 is the inverse of 7 modulo 83, we need to show that when we multiply these numbers together and take the remainder after dividing by 83, the result is 1. For 50 and 5, we calculate 50 x 5 = 250. When dividing 250 by 83, the remainder is 1, confirming that 50 is indeed the inverse of 5 modulo 83. Similarly, for 12 and 7, we calculate 12 x 7 = 84. Dividing 84 by 83 gives a remainder of 1, thus 12 is the inverse of 7 modulo 83.
To find the inverse of 35 modulo 83, we look for a number 'x' such that when we multiply it by 35, the remainder after dividing by 83 is 1. This can be expressed as 35x ≡ 1 (mod 83). Through trial and error or using the Extended Euclidean Algorithm, we find that x = 71 satisfies the equation, making 71 the inverse of 35 modulo 83.
