Question

What is the largest positive integer n such that 1457, 1754, and 368 all leave the same remainder when divided by n?

Answer 1

So we want to solve:$$1457\equiv1754\equiv368\equiv k\mod n$$It follows, then, that:$$1754-1457\equiv k-k\equiv 0\mod n$$so we know that \(1754-1457=297\) is a multiple of \(n\)

Answer 2

similarly we get:$$1457-368=1098\equiv 0\mod n\\1754-368=1386\equiv 0\mod n$$

Answer 3

We want the greatest \(n\) that satisfies the above so it follows it is merely their greatest common divisor \(\gcd(297,1098,1386)\)

Answer 4

So the answer would be 3?

Answer 5

wait but 1457-368 = 1089

Answer 6

good catch :p

Answer 7

clearly \(297=3\times99=3^3\times11,1089=1100-11=11\times99=3^2\times11^2\)

Answer 8

so the answer is 99! oh! I see!

Answer 9

and lastly \(1386=1089+297\) so \(\gcd(297,1089,1386)=\gcd(297,1089)=3^2\times11=99\)

Answer 10

why are you so smart? these problems would have taken me ages to figure out XD

Answer 11

thanks for answering my question! again >.<

Answer 12

haha no I just like these problems a lot ;-p they're fun. no problem!

Answer 13

great! ('cause i have a lot more XD)