Final answer:
To solve the modular equation given an a value of 61, we calculate 3 to the power of 10, apply modulo 9743, and simplify. We thereby find the multiplicative inverse of 16527 modulo 9743 and obtain the solution x ≡ 823.
Step-by-step explanation:
To solve the equation (27*a*x + 131221) mod 9743 = 310, given that a=61, we must first calculate the value of 310 and simplify the equation accordingly.
Firstly, 310 is the same as 31 multiplied by itself 10 times, which equals 59049. However, note that we are dealing with a modular equation, which means we're interested in the remainder when 59049 is divided by 9743. In this case, 310 mod 9743 is 218.
Inserting the given value for 'a' into our starting equation, we get (27*61*x + 131221) mod 9743 = 218. Simplifying, 16527*x + 131221 ≡ 218 mod 9743.
Step by step solution:
- Subtract 131221 from both sides of the congruence: 16527*x ≡ 218 - 131221 mod 9743.
- Calculate the right side: -131003 mod 9743, which simplifies to 3940.
- Now we have 16527*x ≡ 3940 mod 9743. To solve for x, we need to find the multiplicative inverse of 16527 mod 9743.
- Using an Extended Euclidean algorithm or a modular inverse calculator, we get that the multiplicative inverse of 16527 mod 9743 is 4833.
- Multiply both sides by the inverse: x = 3940 * 4833 mod 9743.
- The solution for x is thus x ≡ 823.