Question

Find the greatest number which divides $2011$ and $2623$ leaving remainders $9$ and $5$ respectively.

Answer 1

Let $n$ divides $2011,2623$ leaving the remainder $9,5$ respectively.

The, $2011$ can be expressed as,

$2011=np+9$

$np=2011-9=2002$

$np=2\times 7\times 11\times 13$

And, $2623$ can be expressed as,

$2623=nq+5$

$nq=2623-5=2618$

$nq=2\times 7\times 11\times 17$

But $n$ should have the same value in both instances.

$\therefore n=2\times 7\times 11$

$⟹p=13,q=17$

Thus the number is,

$n=2\times 7\times 11=154$