Answered: "Find the whole number sulotion of the congruence..."

Julieve Vega

Answered question

2022-05-07

Find the whole number sulotion of the congruence equation of (5x+4)=2 mod 8

Jazz Frenia

Skilled2023-05-05Added 106 answers

The given congruence equation is:

5 x + 4 \equiv 2 (\mod 8)

We want to find the whole number solutions for this congruence equation.
First, let's simplify the left-hand side of the congruence equation:

5 x + 4 \equiv 2 (\mod 8)

\Rightarrow 5 x \equiv 2 - 4 (\mod 8)

\Rightarrow 5 x \equiv - 2 (\mod 8)

Next, we need to find the multiplicative inverse of 5 modulo 8, denoted as

5^{- 1}

. This is the number such that

5 \times 5^{- 1} \equiv 1 (\mod 8)

.
To find

5^{- 1}

, we can use the extended Euclidean algorithm as follows:

8 = 5 \times 1 + 3

5 = 3 \times 1 + 2

3 = 2 \times 1 + 1

1 = 3 - 2 \times 1

= 3 - (5 - 3 \times 1) \times 1

= 2 \times 3 - 5

= 2 \times (8 - 5) - 5

= - 5 \times 2 + 8

\Rightarrow 5^{- 1} \equiv - 2 (\mod 8)

Now we can multiply both sides of the congruence equation by

5^{- 1}

5 x \equiv - 2 (\mod 8)

\Rightarrow 5^{- 1} \times 5 x \equiv 5^{- 1} \times - 2 (\mod 8)

\Rightarrow x \equiv - 10 (\mod 8)

\Rightarrow x \equiv 6 (\mod 8)

Therefore, the whole number solution for the congruence equation

5 x + 4 \equiv 2 (\mod 8)

x \equiv 6 (\mod 8)

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