Math Problem Statement
Solution
The problem inThe the question image asks is to:
solve** the" modularFind equation (: x
**)Find such that x \( such that4 ."**
Step-by-Step Solution:
-
The given congruence is:
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Simplify the congruence by dividing through by . However, division in modular arithmetic requires finding the modular inverse of .
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The modular inverse of is a number such that:
4 \$$
p mod Here{,7 we} need \ the). modular**
inverse### of Solution (:
14. \ Simplmodify (7 \4).x Let's \ calculateequiv 4 \ itp:
mod {7} ):
Divide - both sides by 4 (since and are coprime):
for x \equiv 1 \( \pmod{7} b.
Final Answer:
$$x \equiv 1 \p)mod{7}$$.
Would you like such a detailed explanation of modular arithmetic?
Related Questions: that
- What happens if \( a and are not copr4ime in ax \equbiv b \pmod \{m}?
2equ. How can we solve ?
13 .p Whatmod does{ "7mod}ulo "): mean in - arithmetic For? (
b4 =. Can2 a \ modular equation), ( have multiple4 solutions ?cd
5ot. 2 How = is modular8 arithmetic \ usedequ iniv crypt ography1? \
p**modTip{:**7 When} dividing \ in).
modular equations Thus,, ensure the the modular divisor inverse is of cop is .
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Multiply both sides of the congruence by (the modular inverse of ):
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Simplify:
-
Reduce :
Final Answer:
Would you like a detailed explanation of any step or have further questions?
5 Related Questions:
- How is the modular inverse calculated in other cases where ?
- What happens if the coefficients in modular arithmetic are not coprime with the modulus?
- Can you solve ?
- How is modular arithmetic applied in cryptography, such as RSA?
- What is the general rule for solving congruences with multiple solutions?
Tip:
In modular arithmetic, always verify the modulus and check for the modular inverse to simplify division operations efficiently.
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Math Problem Analysis
Mathematical Concepts
Modular Arithmetic
Congruences
Modular Inverse
Formulas
a * x ≡ b mod m
Finding the modular inverse
Theorems
Modular Inverse Theorem
Reduction of congruences
Suitable Grade Level
Grades 9-12