Solve using Linear Congruences and Divisibility.

Question

Let r be the common remainder when 1059, 1417 and 2312 are divided by d>1. Find the value of d-r. Find using linear congruences and divisibility.

Answer 1

$d$ divides $1059-r,1417-r,2312-r$

$\implies d$ divides $1417-r-(1059-r)=358$

and $d$ divides $2312-r-(1417-r)=895$ and $2312-r-(1059-r)=1253$

Hence, $d$ divides $(358,895,1253)=?$

Answer 2

Here you can solve this by using a system of linear Diophantine equation. $$1059≡r\mod d$$ $$1417≡r\mod d$$ $$2312≡r\mod d$$ Hence the differences $$358=2\times 179,$$ $$895=5\times 179,$$ $$1253=7\times 179$$ are divisible by $d.$
I hope you can solve the rest.