+826 What is the largest positive integer $n$ such that $1457$, $1797$, $709$, $15$, $24$, $197$, $428$ all leave the same remainder when divided by $n$?
+1926 Let's focus on the two smallest numbers first. 15 and 24.
Since N leaves a remainder, we can elmiinate the factors. There are
Thus, N cannot be \( 3, 5, 2, 4, 6, 8, 12\)
This leaves 7, 9, 10, 11, 13, and 14. Now, let's test out all the other numbers
15/7 = 2 R 1 24/7 = 3 R 3 so it isn't 7
15/9 = 1 R 6 24/9 = 2 R 6 so 9 looks hopeful, we'll try another
1457/9 = 161 R 8 so 9 is eliminated
15/10 = 1 R 5 24/10 = 2 R 4 so it isn't 10
15/11 = 1 R 4 24/11 = 2 R 2 so it isn't 11
15/13 = 1 R 2 24/13 = 1 R 11 so it isn't 13
15/14 = 1 R 1 24/14 = 1 R 10 so it isn't 14
mmm...I don't think there is a solution
Thanks! :)
