Question
Find the smallest number which when divided by 7, 9 and
12 leaves a remainder 5 in each case.Solution
Let the number be N. N ≡ 5 (mod 7), (mod 9), (mod 12) ⇒ N − 5 is divisible by 7, 9 and 12 LCM(7, 9, 12): 7 = 7 9 = 3² 12 = 2² × 3 LCM = 2² × 3² × 7 = 4 × 9 × 7 = 252 So N − 5 = 252 ⇒ N = 252 + 5 = 257
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