Solved on Mar 27, 2024

Solve the modular linear equation $19x \equiv 4 \pmod{141}$ for $x$ .

STEP 1

1. The expression $19 x \equiv 4 \bmod (141)$ is a linear congruence, which means we are looking for all integers $x$ such that when $19x$ is divided by $141$ , the remainder is $4$ .
2. The linear congruence has a solution if and only if the greatest common divisor (gcd) of $19$ and $141$ divides $4$ .
3. The solution can be found using the Euclidean algorithm to find the modular inverse of $19$ modulo $141$ .

STEP 2

1. Verify that the linear congruence has a solution by checking if the gcd of $19$ and $141$ divides $4$ .
2. Use the Euclidean algorithm to find the gcd of $19$ and $141$ .
3. Use the Extended Euclidean algorithm to find the modular inverse of $19$ modulo $141$ .
4. Multiply the modular inverse by $4$ to find the solution to the congruence.

STEP 3

Check if the gcd of $19$ and $141$ divides $4$ .
$\text{gcd}(19, 141) \,|\, 4$

STEP 4

Use the Euclidean algorithm to find the gcd of $19$ and $141$ .
$\text{gcd}(19, 141)$

STEP 5

Perform the Euclidean algorithm step by step.
$141 = 19 \cdot 7 + 8$ $19 = 8 \cdot 2 + 3$ $8 = 3 \cdot 2 + 2$ $3 = 2 \cdot 1 + 1$ $2 = 1 \cdot 2 + 0$

STEP 6

Identify the last non-zero remainder to find the gcd.
$\text{gcd}(19, 141) = 1$

STEP 7

Since the gcd of $19$ and $141$ is $1$ , and $1$ divides $4$ , the linear congruence has a solution.

STEP 8

Use the Extended Euclidean algorithm to find the modular inverse of $19$ modulo $141$ .
$19x \equiv 1 \bmod (141)$

STEP 9

Work backwards from the Euclidean algorithm to express the gcd as a linear combination of $19$ and $141$ .
$1 = 3 - 2 \cdot 1$ $1 = 3 - (8 - 3 \cdot 2) \cdot 1$ $1 = 3 \cdot 3 - 8 \cdot 1$ $1 = (19 - 8 \cdot 2) \cdot 3 - 8 \cdot 1$ $1 = 19 \cdot 3 - 8 \cdot 7$ $1 = 19 \cdot 3 - (141 - 19 \cdot 7) \cdot 1$ $1 = 19 \cdot 10 - 141 \cdot 1$

STEP 10

Identify the coefficient of $19$ as the modular inverse of $19$ modulo $141$ .
$19 \cdot 10 \equiv 1 \bmod (141)$

STEP 11

Multiply the modular inverse by $4$ to find the solution to the congruence.
$x \equiv 10 \cdot 4 \bmod (141)$

STEP 12

Calculate the product and reduce modulo $141$ if necessary.
$x \equiv 40 \bmod (141)$
The solution to the congruence $19 x \equiv 4 \bmod (141)$ is $x \equiv 40 \bmod (141)$ .

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Solve the modular linear equation $19x \equiv 4 \pmod{141}$ for $x$ .

STEP 1

STEP 2

STEP 3

Check if the gcd of $19$ and $141$ divides $4$ .
$\text{gcd}(19, 141) \,|\, 4$

STEP 4

Use the Euclidean algorithm to find the gcd of $19$ and $141$ .
$\text{gcd}(19, 141)$

STEP 5

Perform the Euclidean algorithm step by step.
$141 = 19 \cdot 7 + 8$ $19 = 8 \cdot 2 + 3$ $8 = 3 \cdot 2 + 2$ $3 = 2 \cdot 1 + 1$ $2 = 1 \cdot 2 + 0$

STEP 6

Identify the last non-zero remainder to find the gcd.
$\text{gcd}(19, 141) = 1$

STEP 7

Since the gcd of $19$ and $141$ is $1$ , and $1$ divides $4$ , the linear congruence has a solution.

STEP 8

Use the Extended Euclidean algorithm to find the modular inverse of $19$ modulo $141$ .
$19x \equiv 1 \bmod (141)$

STEP 9

STEP 10

Identify the coefficient of $19$ as the modular inverse of $19$ modulo $141$ .
$19 \cdot 10 \equiv 1 \bmod (141)$

STEP 11

Multiply the modular inverse by $4$ to find the solution to the congruence.
$x \equiv 10 \cdot 4 \bmod (141)$

STEP 12

Calculate the product and reduce modulo $141$ if necessary.
$x \equiv 40 \bmod (141)$
The solution to the congruence $19 x \equiv 4 \bmod (141)$ is $x \equiv 40 \bmod (141)$ .

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Solve the modular linear equation 19x≡4(mod141)19x \equiv 4 \pmod{141}19x≡4(mod141) for xxx.

STEP 1

STEP 2

STEP 3

Check if the gcd of 191919 and 141141141 divides 444.gcd(19,141) ∣ 4 \text{gcd}(19, 141) \,|\, 4 gcd(19,141)∣4

STEP 4

Use the Euclidean algorithm to find the gcd of 191919 and 141141141.gcd(19,141) \text{gcd}(19, 141) gcd(19,141)

STEP 5

Perform the Euclidean algorithm step by step.141=19⋅7+8 141 = 19 \cdot 7 + 8 141=19⋅7+8 19=8⋅2+3 19 = 8 \cdot 2 + 3 19=8⋅2+3 8=3⋅2+2 8 = 3 \cdot 2 + 2 8=3⋅2+2 3=2⋅1+1 3 = 2 \cdot 1 + 1 3=2⋅1+1 2=1⋅2+0 2 = 1 \cdot 2 + 0 2=1⋅2+0

STEP 6

Identify the last non-zero remainder to find the gcd.gcd(19,141)=1 \text{gcd}(19, 141) = 1 gcd(19,141)=1

STEP 7

Since the gcd of 191919 and 141141141 is 111, and 111 divides 444, the linear congruence has a solution.

STEP 8

Use the Extended Euclidean algorithm to find the modular inverse of 191919 modulo 141141141.19x≡1 mod (141) 19x \equiv 1 \bmod (141) 19x≡1mod(141)

STEP 9

STEP 10

Identify the coefficient of 191919 as the modular inverse of 191919 modulo 141141141.19⋅10≡1 mod (141) 19 \cdot 10 \equiv 1 \bmod (141) 19⋅10≡1mod(141)

STEP 11

Multiply the modular inverse by 444 to find the solution to the congruence.x≡10⋅4 mod (141) x \equiv 10 \cdot 4 \bmod (141) x≡10⋅4mod(141)

STEP 12

Calculate the product and reduce modulo 141141141 if necessary.x≡40 mod (141) x \equiv 40 \bmod (141) x≡40mod(141)The solution to the congruence 19x≡4 mod (141)19 x \equiv 4 \bmod (141)19x≡4mod(141) is x≡40 mod (141)x \equiv 40 \bmod (141)x≡40mod(141).

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Solve the modular linear equation $19x \equiv 4 \pmod{141}$ for $x$ .

Check if the gcd of $19$ and $141$ divides $4$ .
$\text{gcd}(19, 141) \,|\, 4$

Use the Euclidean algorithm to find the gcd of $19$ and $141$ .
$\text{gcd}(19, 141)$

Perform the Euclidean algorithm step by step.
$141 = 19 \cdot 7 + 8$ $19 = 8 \cdot 2 + 3$ $8 = 3 \cdot 2 + 2$ $3 = 2 \cdot 1 + 1$ $2 = 1 \cdot 2 + 0$

Identify the last non-zero remainder to find the gcd.
$\text{gcd}(19, 141) = 1$

Since the gcd of $19$ and $141$ is $1$ , and $1$ divides $4$ , the linear congruence has a solution.

Use the Extended Euclidean algorithm to find the modular inverse of $19$ modulo $141$ .
$19x \equiv 1 \bmod (141)$

Identify the coefficient of $19$ as the modular inverse of $19$ modulo $141$ .
$19 \cdot 10 \equiv 1 \bmod (141)$

Multiply the modular inverse by $4$ to find the solution to the congruence.
$x \equiv 10 \cdot 4 \bmod (141)$

Calculate the product and reduce modulo $141$ if necessary.
$x \equiv 40 \bmod (141)$
The solution to the congruence $19 x \equiv 4 \bmod (141)$ is $x \equiv 40 \bmod (141)$ .