Consider the linear congruence 6 x andequiv; 3 (mod 9). Then the incongruent solutions modulo 9 of this congruence are:a)0, 3, 6b)no solutionc)2, 5, 8d)1, 4, 7Correct answer is option 'C'. Can you explain this answer?

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Consider the linear congruence 6 x ≡ 3 (mod 9). Then the incongruent solutions modulo 9 of this congruence are:

a)
0, 3, 6
b)
no solution
c)
2, 5, 8
d)
1, 4, 7

Correct answer is option 'C'. Can you explain this answer?

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Verified Answer

Consider the linear congruence 6 x ≡ 3 (mod 9). Then the incongr...

Concept:
Given: ax ≡ b (mod m) --(1)
Step 1:
Find d,
d = the GCD of (a,m)
We have exactly 'd', incongruent solutions
Step 2:
Divide (1) with 'd', on both sides

find x₁ as one solution of this congruency,
Step 3:
Now, find the incongruent solutions,
x = x₁ + m/dp -----(2)
Where, 0 ≤ p ≤ (d-1)
Putting the values of 'p' in (2), we will get the required solutions.
Analysis:
6 x ≡ 3 (mod 9) ---(3)
d = GCD(6, 9) = 3
We have exactly 3 incongruent solutions,
Now, dividing (3), with 3, we get
⇒ 6x/3 ≡ 33 (mod 9/3)
2x ≡ 1 (mod 3)
We can manually see that x = 2,
4 - 1 ≡ 3 and (3 mod 3) = 0
One solution is x₁ = 2,
Now, to find all the incongruent solutions,
x = x₁ + m/d p
Where, 0 ≤ p ≤ (3-1)
put, p = 0,
x = 2 + 0 = 2
put, p = 1,
x = 2 + 3 × 1 = 5
put, p = 2,
x = 2 + 3 × 2 = 8
The three incongruent solutions modulo 9 of this congruence are: 2, 5, 8

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Most Upvoted Answer

Consider the linear congruence 6 x ≡ 3 (mod 9). Then the incongr...

Concept:
Given: ax ≡ b (mod m) --(1)
Step 1:
Find d,
d = the GCD of (a,m)
We have exactly 'd', incongruent solutions
Step 2:
Divide (1) with 'd', on both sides

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Community Answer

Consider the linear congruence 6 x ≡ 3 (mod 9). Then the incongr...

Explanation:

Linear Congruence:
- The linear congruence given is 6x ≡ 3 (mod 9).
- This can be rewritten as 6x ≡ 3 (mod 9) or 6x ≡ 3 + 9k, where k is an integer.

Finding Solutions:
- To find the incongruent solutions modulo 9, we need to solve the congruence equation.
- We can simplify the equation by dividing both sides by 3: 2x ≡ 1 (mod 3).
- The solutions to this simplified congruence are x ≡ 2, x ≡ 5, and x ≡ 8 (mod 9).

Substitute Back:
- To find the solutions of the original congruence, we substitute the values of x back into the equation.
- Substituting x = 2, we get 6(2) ≡ 12 ≡ 3 (mod 9), which satisfies the congruence.
- Substituting x = 5, we get 6(5) ≡ 30 ≡ 3 (mod 9), which also satisfies the congruence.
- Substituting x = 8, we get 6(8) ≡ 48 ≡ 3 (mod 9), satisfying the congruence as well.

Final Answer:
- The incongruent solutions modulo 9 of the congruence 6x ≡ 3 (mod 9) are x ≡ 2, 5, 8 (mod 9), which are 2, 5, and 8.

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Consider the linear congruence 6 x ≡ 3 (mod 9). Then the incongruent solutions modulo 9 of this congruence are:a)0, 3, 6b)no solutionc)2, 5, 8d)1, 4, 7Correct answer is option 'C'. Can you explain this answer?

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Consider the linear congruence 6 x ≡ 3 (mod 9). Then the incongruent solutions modulo 9 of this congruence are:a)0, 3, 6b)no solutionc)2, 5, 8d)1, 4, 7Correct answer is option 'C'. Can you explain this answer? for SAT 2025 is part of SAT preparation. The Question and answers have been prepared according to the SAT exam syllabus. Information about Consider the linear congruence 6 x ≡ 3 (mod 9). Then the incongruent solutions modulo 9 of this congruence are:a)0, 3, 6b)no solutionc)2, 5, 8d)1, 4, 7Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for SAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the linear congruence 6 x ≡ 3 (mod 9). Then the incongruent solutions modulo 9 of this congruence are:a)0, 3, 6b)no solutionc)2, 5, 8d)1, 4, 7Correct answer is option 'C'. Can you explain this answer?.

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