Can you do #6? ## Polynomial Congruences### Exercises1. Solve the following polynomial congruences: - (a) \( x^2 + 12x - 17 \equiv 0 \pmod{143} \) - (b) \( x^3 + 30x^2 + 27x + 23 \equiv 0 \pmod{45} \) 2. Determine, without using a computer, all integers \( x \) such that the last three digits of \( x^3 \) are the same as those of \( x \).3. Let \( m = p_1^{a_1} \cdots p_k^{a_k} \). Suppose the congruence \( f(x) \equiv 0 \pmod{p_i^{a_i}} \) has \( n_i \) distinct solutions for \( 1 \leq i \leq k \). Show that \( f(x) \equiv 0 \pmod{m} \) has \( n_1 n_2 \cdots n_k \) distinct solutions.4. Let \( p \) be an odd prime. Show that \( x^2 \equiv a \pmod{p^k} \), \( k \geq 2 \) has exactly two solutions if \( x^2 \equiv a \pmod{p} \) has two solutions, and no solutions if \( x^2 \equiv a \pmod{p} \) has no solutions.5. This exercise determines the solutions to \( x^2 \equiv 1 \pmod{2^k} \), \( k \geq 1 \). - (a) Use a computer to determine the solutions for \( k = 1, 2, \ldots, 10 \). Do you see a general pattern? - (b) Use the technique of Theorem 3.4.6 to construct the solutions to \( x^2 \equiv 1 \pmod{16} \) from the set of solutions to \( x^2 \equiv 1 \pmod{8} \). Which solutions of \( x^2 \equiv 1 \pmod{8} \) give solutions to \( x^2 \equiv 1 \pmod{16} \) and why?

Name: Can you do #6? ## Polynomial Congruences### Exercises1. Solve the foll
Uploaded: 2024-03-20T07:06:41.000Z
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Description: Can you do #6? ## Polynomial Congruences### Exercises1. Solve the following polynomial congruences: - (a) \( x^2 + 12x - 17 \equiv 0 \pmod{143} \) - (b) \( x^3 + 30x^2 + 27x + 23 \equiv 0 \pmod{45} \) 2. Determine, without using a computer, all