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Chapter 10: Problem 42

Show that \((x-9)^{4}\) is not congruent to \(\left(x^{4}-9\right)\) modulo 4.

Short Answer

Expert verified
It has been confirmed that \((x-9)^{4}\) is not congruent to \(\left(x^{4}-9\right)\) modulo 4 since their modulo 4 forms are \(x^4 + 2x^2 + 1(mod 4)\) and \(x^{4}-1(mod 4)\) respectively, and these are not equal.

Step by step solution

01

Expanding \((x-9)^{4}\)

Using the binomial theorem, expand \((x-9)^{4}\) to get \(x^4 - 4x^3*9 + 6x^2*81 -4x*729 + 6561\).
02

Applying modulo operation on \((x-9)^{4}\)

Take each term of the resultant expression modulo 4. Doing this operation, you get \(x^4 - 0 + 2x^2*1- 0 + 1 = x^4 + 2x^2 + 1 (mod 4)\).
03

Applying modulo operation on \(\left(x^{4}-9\right)\)

Take \(\left(x^{4}-9\right)\) modulo 4. You get: \(x^{4}-1\) (because 9 modulo 4 equals to 1).
04

Comparing the results

Compare the two results: \(x^4 + 2x^2 + 1(mod 4)\) and \(x^{4}-1(mod 4)\). You see that the two are not the same, so \((x-9)^{4}\) is not congruent to \(\left(x^{4}-9\right)\) modulo 4.

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