Question
Find the least positive integer N such that N ≡ 3 (mod
7), N ≡ 4 (mod 9), and N ≡ 5 (mod 8).Solution
ATQ,
From mod 8: N = 8k + 5. Plug into mod 7: 8k + 5 ≡ k + 5 ≡ 3 (mod 7) ⇒ k ≡ −2 ≡ 5 (mod 7). So k = 7t + 5 ⇒ N = 8(7t + 5) + 5 = 56t + 45. Now enforce mod 9: 56t + 45 ≡ 2t + 0 (mod 9) since 56≡2, 45≡0. So 2t ≡ 4 (mod 9) ⇒ t ≡ 2 (mod 9). ⇒ t = 9s + 2. Least positive: s=0 ⇒ t=2 ⇒ N = 56·2 + 45 = 157.
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