Question
Find the smallest positive integer which when divided by
5, 7 and 9 leaves a remainder 4 in each case.Solution
If N leaves remainder 4 when divided by 5, 7, 9, then N − 4 is divisible by all 5, 7, 9. LCM(5,7,9): 5 = 5 7 = 7 9 = 3² LCM = 5 × 7 × 3² = 5 × 7 × 9 = 315 So N − 4 = 315k ⇒ N = 315k + 4 Smallest positive N when k = 1: N = 315 × 1 + 4 = 319 Check: 319 ÷ 5 → remainder 4 319 ÷ 7 → 7×45 = 315, remainder 4 319 ÷ 9 → 9×35 = 315, remainder 4 Answer: 319.
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