Question

Find the largest number by which when $245$and $1029$ are divided, remainder in each case is $5$.

Answer 1

The largest number which divides $245$ and $1029$, gets remainder $5$.

The number which is exactly divisible is $245-5=240$ and $1029-5=1024$

Prime factorization of $240$ and $1024$ are

$$\begin{gathered} 240=2\times 2\times 2\times 2\times 3\times 5 \\ 1024=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2 \end{gathered}$$

Common factors are $2,2,2,2$

H.C.F of $240$ and $1024$ is $2\times 2\times 2\times 2=16$

Hence, the largest number by which when $245$and $1029$ are divided and gives remainder $5$ is $16.$