Math Problem Statement
For which of the following 3-tuples of integers (a,b,c) do we have a solution to the system of simultaneous linear congruences: n≡a(mod4) , n≡b(mod7) , n≡c(mod28) ?
(1,6,6)
(0,1,8)
(1,1,9)
(1,2,14)
(2,4,18)
(1,2,23)
Solution
To determine which of the given 3-tuples have a solution to the system of simultaneous linear congruences:
we need to check whether a value of exists that satisfies all three congruences simultaneously for each tuple .
Relationship Between the Congruences
- implies and .
- Therefore, for a solution to exist, must be congruent to , and must be congruent to .
Checking Each Tuple
-
Tuple :
- Check ? No.
- Check ? Yes.
No solution exists since is not congruent to .
-
Tuple :
- Check ? Yes.
- Check ? No.
No solution exists since is not congruent to .
-
Tuple :
- Check ? Yes.
- Check ? Yes.
A solution exists since both conditions are satisfied.
-
Tuple :
- Check ? No.
- Check ? Yes.
No solution exists since is not congruent to .
-
Tuple :
- Check ? Yes.
- Check ? Yes.
A solution exists since both conditions are satisfied.
-
Tuple :
- Check ? No.
- Check ? Yes.
No solution exists since is not congruent to .
Conclusion:
The 3-tuples and are the ones for which a solution to the system of simultaneous linear congruences exists.
Would you like more details, or do you have any questions?
Here are 5 follow-up questions:
- What is the importance of the moduli in determining the consistency of congruences?
- How can the Chinese Remainder Theorem help when moduli are not pairwise coprime?
- What happens if the moduli are not coprime in a system of linear congruences?
- How can systems of linear congruences be applied in cryptography?
- What are the steps to check for a solution manually when the moduli are large?
Tip: Always check the congruences modulo smaller factors before dealing with larger composite moduli. This simplifies the problem and makes it easier to spot inconsistencies.
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Math Problem Analysis
Mathematical Concepts
Modular arithmetic
Linear congruences
Chinese Remainder Theorem
Formulas
-
Theorems
Chinese Remainder Theorem
Suitable Grade Level
Advanced High School and above
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