Step 1

Here, we have to solve for congruences

2x=1 mod 5

x=3 mod 4

To solve such questions, we apply chinese remainder theorem.

Chinese Remainder Theorem- Let m, n are relatively prime integers. Then the system of simultaneous congruences \(\displaystyle{x}\equiv{a}_{{1}}{\left(\text{mod}{m}\right)}{x}\equiv{a}_{{2}}{\left(\text{mod}{n}\right)}\) has a unique solution modulo M = mxn, for any given integers \(\displaystyle{a}_{{1}},{a}_{{2}}\).

Step 2

\(\displaystyle{2}{x}={1}\text{mod}{5}\Rightarrow{2}\times{3}={3}\text{mod}{5}\Rightarrow{x}={3}\text{mod}{5}\)

and we have x = 3 mod 4

Here, 4 and 5 are co-prime, therefore By chinese remainder theorem, it has unique solution module 20.

So, only possibilty is x=23 which satisfy the given congruences.

Step 3

Answer:

x=23

Here, we have to solve for congruences

2x=1 mod 5

x=3 mod 4

To solve such questions, we apply chinese remainder theorem.

Chinese Remainder Theorem- Let m, n are relatively prime integers. Then the system of simultaneous congruences \(\displaystyle{x}\equiv{a}_{{1}}{\left(\text{mod}{m}\right)}{x}\equiv{a}_{{2}}{\left(\text{mod}{n}\right)}\) has a unique solution modulo M = mxn, for any given integers \(\displaystyle{a}_{{1}},{a}_{{2}}\).

Step 2

\(\displaystyle{2}{x}={1}\text{mod}{5}\Rightarrow{2}\times{3}={3}\text{mod}{5}\Rightarrow{x}={3}\text{mod}{5}\)

and we have x = 3 mod 4

Here, 4 and 5 are co-prime, therefore By chinese remainder theorem, it has unique solution module 20.

So, only possibilty is x=23 which satisfy the given congruences.

Step 3

Answer:

x=23