Consider the fourteen numbers, 14, 24 ,….,144. The smallest natural number $n$ such that they leave distinct remainders when divided by $n$ is:

Step 1: 14, 24 ……..,144

Step 2: x4  $a$ (mod n)

Step 3: y4  $b$ (mod n) such that $a$ ≠ $b$ for $x$ ≠ $y$ and x, $y$ in 1, 2, \dots.14

Step 4: ( $x$ − $y$ )  (a − $b$ ) (mod n)

Step 5: => ( $x$ − $y$ ) ( $x$ + $y$ ) ( $x$ 2 + $y$ 2 )  ( $a$ − $b$ ) (mod n)

Step 6: => (

Step 7: n | ( $x$ − $y$ )( $x$ + $y$ ) $x$ 2 + $y$ 2 ) …(i)

Step 8: We have to find minimum $n$ with condition (i)

Step 9: Clearly, $n$ > 27 as ( $x$ + $y$ ) in 3, \dots 27

Step 10: Now $n$ = 28, $x$ = 6, $y$ = 8 works

Step 11: n = 29, $x$ = 5, $y$ = 2 works

Step 12: n = 30, $x$ = 8, $y$ = 2 works

Step 13: for $x$ = 31, there are no such x, y,

Step 14: 31| ( $x$ − $y$ )( $x$ + $y$ ) $x$ 2 + $y$ 2 )

Step 15: Must be prime factor

Step 16: ( )

Step 17: 31| $x$ 2 + $y$ 2 and 31| ( $x$ − $y$ )( $x$ + $y$ )

Step 18: => 31 will be the answer