Find the smallest nonnegative integer for x such that x ≡ 1...

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Find the smallest nonnegative integer for x such that x ≡ 1 (mod 2), x ≡ 2 (mod 3), x ≡ 3 (mod 5), and x ≡ 4 (mod 11). Solution: To find the smallest nonnegative integer that satisfies all the given congruences, we can use the Chinese Remainder Theorem (CRT). First, let's solve the congruences individually: 1. x ≡ 1 (mod 2) The solutions for this congruence are x = 1, 3, 5, 7, 9, 11, ... 2. x ≡ 2 (mod 3) The solutions for this congruence are x = 2, 5, 8, 11, 14, ... 3. x ≡ 3 (mod 5) The solutions for this congruence are x = 3, 8, 13, 18, 23, ... 4. x ≡ 4 (mod 11) The solutions for this congruence are x = 4, 15, 26, 37, 48, ... Now, let's find the smallest nonnegative integer that satisfies all the congruences. By observing the solutions, we can see that the smallest nonnegative integer that satisfies all the congruences is x = 8. Therefore, the smallest nonnegative integer for x that satisfies all the given congruences is x = 8.

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Question

Find the smallest nonnegative integer for x such that $x \equiv 1 \pmod{2}$ $x \equiv 2 \pmod{3}$ $x \equiv 3 \pmod{5}$ $x \equiv 4 \pmod{11}$

Question

Find the smallest nonnegative integer for x such that $x \equiv 1 \pmod{2}$ $x \equiv 2 \pmod{3}$ $x \equiv 3 \pmod{5}$ $x \equiv 4 \pmod{11}$

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