How we can find number $b$ which is divisor of two divison

Question

We divided two number $83$ and $107$ by a natural number $b$.

Remainder of first division is $5$ ($83$ by $b$) and remainder of second division is $3$ ($107$ by $b$).

How we can find number $b$ ?

Answer 1

Hint:

$b\mid 83-5=78$ and $b\mid107-3=104$ so that:

$$b\mid\gcd(78,104)$$

Answer 2

$83 \equiv 5$ (mod $b$), $107 \equiv 3$ (mod $b$). So $b$ divides $78 = 2 \times 13 \times 3$ and $104 = 2^3 \times 13$. Therefore $b$ divides $2 \times 13 = 26$, so $1, 2, 13$, or $26$ are possibilities for $b$. Either $b = 13$ or $b = 26$ are okay, but we exclude $b = 1, 2$ because the remainders are too big.