When positive integer n is divided by 5, the remainder is 1, when n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 35?

yellowho wrote:When positive integer n is divided by 5, the remainder is 1, when n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 35? The answer would be 4 . n = 31 k = 4 n + k = 35 n for 5 giving remainder 1 = [6,11,16,21,26,31,36,41,46,51] n for 7 giving remainder 3 = [10,17,24,31,38,45,52,59,66,73] Smallest number common between both sets is 31 . 31/5 gives remainder 1 and 31/7 gives remainder 3 .

yeah. my way: (testing number) 0 is the first multiple of 35; can't come up with a k and n to satisfy next 35, works. I guess we got lucky here is there a "better" way to do this? Perhaps the experts on modulus can chime in.

k+n is a multiple of 35 - so it an be 35,70 etc if it is 35, then n shud be 31 to satify the remainders of both 5 and 7. 31+4=35, so k=4 same wth 70. n shud be 66, here too k=4. this is how i did it. ya but the 0 one doesnt work out though. yellowho wrote:yeah. my way: (testing number) 0 is the first multiple of 35; can't come up with a k and n to satisfy next 35, works. I guess we got lucky here is there a "better" way to do this? Perhaps the experts on modulus can chime in.

yellowho wrote:When positive integer n is divided by 5, the remainder is 1, when n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 35? Just note that (n + 4) is multiple of both 5 and 7. Smallest possible value of (n + 4) is 35 (zero is discarded as that would make n negative) Therefore, smallest possible value of k is nothing but 4.

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