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Chinese Remainder Theorem Calculator calculator

Solve two or three congruences with pairwise coprime positive moduli and return the least nonnegative solution.

Chinese Remainder Theorem Calculator calculator

Resultx ≡ 23 (mod 105)

What this calculator answers

Chinese Remainder Theorem Calculator calculator applies the registered method “solve two or three pairwise-coprime congruences and normalize the unique result modulo the product of their moduli” to the displayed Number of congruences, Residue 1, Modulus 1, Residue 2, Modulus 2, Residue 3, Modulus 3 inputs. Both worked examples come from the production calculation engine, and NIST Digital Library of Mathematical Functions — Chinese remainder theorem defines the cited method or convention; no hidden inputs or current external data are inferred.

Variables, defaults, and limits

Number of congruences
Number of congruences is a choice input. The displayed starter value is Three congruences; it is an example, not a hidden assumption.
Default: Three congruences
Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result. Available choices: Two congruences, Three congruences.
Residue 1
Residue 1 is a structured text input. The displayed starter value is 2; it is an example, not a hidden assumption.
Default: 2
Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result.
Modulus 1
Modulus 1 is a structured text input. The displayed starter value is 3; it is an example, not a hidden assumption.
Default: 3
Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result.
Residue 2
Residue 2 is a structured text input. The displayed starter value is 3; it is an example, not a hidden assumption.
Default: 3
Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result.
Modulus 2
Modulus 2 is a structured text input. The displayed starter value is 5; it is an example, not a hidden assumption.
Default: 5
Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result.
Residue 3
Residue 3 is a structured text input. The displayed starter value is 2; it is an example, not a hidden assumption.
Default: 2
Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result.
Modulus 3
Modulus 3 is a structured text input. The displayed starter value is 7; it is an example, not a hidden assumption.
Default: 7
Accepted values: The field has no narrower HTML limit, but it must still satisfy the documented formula domain and produce a finite result.

Formula and calculation rule

R≡x₂ (mod x₃); R≡x₄ (mod x₅); [R≡x₆ (mod x₇) iff x₁=threeCongruences]; x₁=Number of congruences; x₂=Residue 1; x₃=Modulus 1; x₄=Residue 2; x₅=Modulus 2; x₆=Residue 3; x₇=Modulus 3

Chinese Remainder Theorem Calculator calculator evaluates the registered expression “R≡x₂ (mod x₃); R≡x₄ (mod x₅); [R≡x₆ (mod x₇) iff x₁=threeCongruences]; x₁=Number of congruences; x₂=Residue 1; x₃=Modulus 1; x₄=Residue 2; x₅=Modulus 2; x₆=Residue 3; x₇=Modulus 3” with the validated inputs. Full calculation precision is retained until the result is formatted for display.

Worked examples

Chinese Remainder Theorem Calculator calculator evaluated example 1

Use the page’s labelled starter inputs to verify Chinese Remainder Theorem Calculator calculator.

Inputs
  • Number of congruences: Three congruences
  • Residue 1: 2
  • Modulus 1: 3
  • Residue 2: 3
  • Modulus 2: 5
  • Residue 3: 2
  • Modulus 3: 7

Evaluated result: x ≡ 23 (mod 105)

Chinese Remainder Theorem Calculator calculator evaluated example 2

Change one valid input or choice and evaluate Chinese Remainder Theorem Calculator calculator again to check that the result responds deterministically.

Inputs
  • Number of congruences: Three congruences
  • Residue 1: 3
  • Modulus 1: 3
  • Residue 2: 3
  • Modulus 2: 5
  • Residue 3: 2
  • Modulus 3: 7

Evaluated result: x ≡ 93 (mod 105)

Assumptions

  • Chinese Remainder Theorem Calculator calculator uses the visitor-entered values exactly as labelled; it does not retrieve private records or current rates.
  • The calculation is limited to the declared relationship: R≡x₂ (mod x₃); R≡x₄ (mod x₅); [R≡x₆ (mod x₇) iff x₁=threeCongruences]; x₁=Number of congruences; x₂=Residue 1; x₃=Modulus 1; x₄=Residue 2; x₅=Modulus 2; x₆=Residue 3; x₇=Modulus 3.
  • Intermediate values are not rounded; display rounding is applied only at the presentation boundary.
  • The deterministic calculation runs locally and does not upload calculator inputs to the application API.
  • The result is an educational arithmetic result and excludes facts that are not represented by an input.

Validation and boundaries

  • Every visible required input must be present; a missing value is never replaced with zero.
  • NaN, positive or negative infinity, unsafe overflow, and a non-finite final result are rejected.
  • Field-specific minimums, maximums, and choices apply to Number of congruences, Residue 1, Modulus 1, Residue 2, Modulus 2, Residue 3, Modulus 3.
  • Zero, negative values, and discrete counts are accepted only when the displayed field definition permits them.

Review and correction links

Common mistakes

  • Confirm the meaning and unit of Number of congruences, Residue 1, Modulus 1, Residue 2, Modulus 2, Residue 3, Modulus 3 before calculating; a numerically valid value can still use the wrong convention.
  • Do not round intermediate values when checking the result, because early rounding can change the last displayed digits.
  • Changing an unstated real-world assumption does not change the calculator until the corresponding displayed input is changed.
  • Use the result as the answer to the displayed mathematical question, not to a different word problem with hidden conditions.

Frequently asked questions

What question does Chinese Remainder Theorem Calculator calculator answer?
Solve two or three congruences with pairwise coprime positive moduli and return the least nonnegative solution. The implemented relationship is R≡x₂ (mod x₃); R≡x₄ (mod x₅); [R≡x₆ (mod x₇) iff x₁=threeCongruences]; x₁=Number of congruences; x₂=Residue 1; x₃=Modulus 1; x₄=Residue 2; x₅=Modulus 2; x₆=Residue 3; x₇=Modulus 3.
Which inputs does Chinese Remainder Theorem Calculator calculator use?
It uses the visible fields Number of congruences, Residue 1, Modulus 1, Residue 2, Modulus 2, Residue 3, Modulus 3. No hidden value is substituted for an omitted required input.
What happens when Chinese Remainder Theorem Calculator calculator receives an invalid or extreme value?
The page rejects missing, ambiguous, out-of-range, or non-finite values and refuses to display a non-finite result.
Why can Chinese Remainder Theorem Calculator calculator differ from another result?
Different unit conventions, endpoint policies, fee or rate assumptions, formula domains, and premature rounding can produce a different answer.
Can I use Chinese Remainder Theorem Calculator calculator as professional advice?
No. It answers the displayed mathematical question and cannot account for omitted real-world facts.
How can I verify the Chinese Remainder Theorem Calculator calculator result?
Recalculate the two evaluated examples without rounding intermediate values and compare the declared formula or convention with NIST Digital Library of Mathematical Functions — Chinese remainder theorem.

Source and review scope

NIST Digital Library of Mathematical Functions — Chinese remainder theorem

Scope: NIST Digital Library of Mathematical Functions — Chinese remainder theorem is used to check the formula, definition, or convention relevant to Chinese Remainder Theorem Calculator calculator. The citation does not supply current personal, lender, tax, medical, or market data.

Source checked:

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