Question

What is the largest number that divides $626,3127$ and $15628$ and leaves remainders of $1,2$ and $3$ respectively.

Answer 1

Clearly, the required number is the HC.F of the numbers

$626-1=625,3127-2=3125$ and $15628-3=15625.$

Using Euclid's division lemma to find the H.C.F. of $625$ and $3125$.
$3125=625$ $\times$ $5+0$

Clearly, H.C.F. of $625$ and $3125$ is $625$.

Now, H.C.F. of $625$ and $15625$
$15625=625$ $\times$ $25+0$

So, the H.C.F of $625$ and $15625$ is $625$
Hence, H.C.F of $625,3125$ and $15625$ is $625$.

Hence, the required number is $625$.