Math Problem Statement

Find x such that 4x ≡ 4 mod 7.

Solution

The problem inThe the question image asks is to:

solve** the" modularFind equation (: x

**)Find such that x \( such that4 x4\xequ\iv4mod7x4 \xequ \iv 4 \mod 7."**

Step-by-Step Solution:

  1. The given congruence is: 4x4(mod7).4x \equiv 4 \pmod{7}.

  2. Simplify the congruence by dividing through by 44. However, division in modular arithmetic requires finding the modular inverse of 4mod74 \mod 7.

  3. The modular inverse of amodma \mod m is a number bb 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 Test4Test4 and values7values7 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

  1. What happens if \( a and mm are not copr4ime in ax \equbiv b \pmod \{m}?
    2equ. How can we solve 4x2(modiv7)4x \equiv 2 \pmod{iv7}?
    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 rime4to themodulusmod.7r ime4 to \ the modulusmod .7 is 22.

  1. Multiply both sides of the congruence 4x4(mod7)4x \equiv 4 \pmod{7} by 22 (the modular inverse of 44): 24x24(mod7).2 \cdot 4x \equiv 2 \cdot 4 \pmod{7}.

  2. Simplify: x8(mod7).x \equiv 8 \pmod{7}.

  3. Reduce 8mod78 \mod 7: x1(mod7).x \equiv 1 \pmod{7}.

Final Answer:

x1(mod7).x \equiv 1 \pmod{7}.

Would you like a detailed explanation of any step or have further questions?


5 Related Questions:

  1. How is the modular inverse calculated in other cases where gcd(a,m)>1\gcd(a, m) > 1?
  2. What happens if the coefficients in modular arithmetic are not coprime with the modulus?
  3. Can you solve 5x3mod115x \equiv 3 \mod 11?
  4. How is modular arithmetic applied in cryptography, such as RSA?
  5. 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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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