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What is the largest positive integer $n$ such that $1457$, $1797$, $709$, $15$, $24$, $197$, $428$ all leave the same remainder when divided by $n$?

 Jun 28, 2024
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What is the largest positive integer $n$ such that $1457$, $1797$, $709$, $15$, $24$, $197$, $428$ all leave the same remainder when divided by $n$?   

 

First, let's eliminate as many divisors as we can.  Consider 15 and 24.   

The divisor n has to leave a remainder, so factors can be eliminated.   

 

n cannot be 3 or 5, nor 2, 4, 6, 8, or 12 .... leaving 7, 9, 10, 11, 13, and 14.   

 

Remember, the remainder has to be the same.  Try them out on 15 and 24 first.  

 

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   

 

Well, I have run out of divisors and ideas at the same time.   

Either the problem has no solution, or I'm missing something.   

.

 Jun 29, 2024

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