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

Badada Apr 16, 2019

#1**+1 **

n=1; a=(1457 % n);b=(368%n);c=(1754 %n);if(a==b and b==c, goto5,goto6);printa,b,c,n; n++; if(n<5000, goto1, discard=0;

Rem n

0 0 0 1

2 2 2 3

8 8 8 9

5 5 5 11

5 5 5 33

**71 71 71 99**

**So, the largest n = 99 and the remainder of 71.**

Guest Apr 16, 2019

#2**+2 **

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

heureka Apr 17, 2019

#3**+1 **

Finding the absolute differences of every possible 2-permutation of these 3 integers and writing out its prime factorisation:

|1457 - 368| = 1089 = 3^2 * 11^2

|368 - 1754| = 1386 = 2 * 3^2 * 7 * 11

|1754 - 1457| = 297 = 3^3 * 11

Finding the G.C.D of these absolute differeces:

GCD(3^2 * 11^2, 2*3^2*7*11, 3^3*11) = 3^2 * 11 = **99**

Therefore 99 is the largest positive integer such that 1457, 368, and 1754 leave the same remainder when divided by it.

(Why does this work? The proof is trivial and is left for the reader as an exercise. )

MaxWong Apr 18, 2019