427. If the numbers 826 and 4373 are divided by the same natural number, the remainders will be 7 and 8, respectively. Find all values of the divisor.
$\triangle$ Let's write the corresponding equalities:

$$
826=b q_{1}+7, \quad 4373=b q_{2}+8
$$

where $b-$ is the unknown divisor, $q_{1}$ and $q_{2}$ are the incomplete quotients. Then

$$
b q_{1}=819, \quad b q_{2}=4365
$$

Let's factorize the numbers 819 and 4365 into prime factors:

$$
819=3^{2} \cdot 7 \cdot 13, \quad 4365=3^{2} \cdot 5 \cdot 97
$$

Therefore, the common divisors of the numbers 819 and 4365 are the numbers 1, 3, and 9. But common divisors equal to 1 and 3 are impossible, as the remainder must be less than the divisor. The only remaining number is 9.

Answer: 9.