Question

Find the general form of all positive integers which divided by $5,7,8$ leave remainders $3,2,5$ respectively.

Answer 1

when a positive integer $x$ is divided by $5,6,7,$ and

$8,$ the remainder is $3,2,5$ respectively

$x=5p+3$ ( $x$ could be $3,8,13,23,...93,...$)

$x=7q+2$ ( $x$ could be $2,9,16,...93,...$)

$x=8r+5$ ( $x$ could be $5,13,...93,...$)

Divisor will be at least the LCM of $5,7\&8$

$=280$

And remainder will be the first common integer of three pattern. hence $93$

$\therefore$ the general form can be written as

$x=280t+93$