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Chapter 10: Problem 43

Show that \((x-5)^{3}\) is congruent to \(\left(x^{3}-5\right)\) modulo 3.

Short Answer

Expert verified
By expanding and simplifying the left side of the equation and applying modulo operation, we derived that \(x^3 - 2\) is congruent to \(x^{3}-5\) modulo 3, which shows that \((x-5)^{3}\) is indeed congruent to \(\left(x^{3}-5\right)\) modulo 3.

Step by step solution

01

Expand the left side

Start by expanding the left-side expression \((x-5)^3\). The binomial formula yields: \(x^3 - 15x^2 + 75x - 125\).
02

Apply the modulus operation

Next, apply modulo 3 to each term separately: \(x^3 \mod 3 - 15x^2 \mod 3 + 75x \mod 3 - 125 \mod 3\). The remainder when \(x^3\), \(15x^2\), and \(75x\) are divided by 3 are \(x^3\), \(0\), and \(0\), respectively. The remainder when 125 is divided by 3 is 2, so we are then left with \(x^3 - 2\).
03

Simplify the expression

After simplifying, we are left with \(x^3 - 2\). However, because we are working with modulo 3, \(-2 \mod 3\) is equivalent to \(1 \mod 3\). Thus, our simplified expression matches the right side of the original problem: \(x^3 - 5 \mod 3 = x^3 - 2 = x^3 + 1\) as needed.

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