Find the largest number which divides $853$ and $385$ leaving the remainder $7$ in each case.

Hint: In this question, we have to find the largest number, therefore it means that we have to find the highest common factor (HCF) or the greatest common factor of the two given numbers. So we will first subtract the given remainder from both the numbers and after that we will find the Highest common factor of both the numbers by the prime factorization method. This will give us our answer.

Complete step by step solution:
Here we have two numbers $853$ and $385$ .
We have to find a number which divides $853$ and $385$ with a remainder of $7$. This means, if we subtract $7$ from $853$ and $385$, then the resulting numbers will be exactly divisible by some common numbers. We will pick the largest of those common numbers to get the required result.
Let us subtract $7$ from both the numbers, so it gives us two new numbers i.e.
$853 - 7 = 846$
And the other number is
$385 - 7 = 378$
Here we will now use the prime factorization method to find the HCF of these numbers.
The prime factors $846$ are:
$ \Rightarrow 846 = \;2 \times 3 \times 3 \times 3 \times 47$
Now the prime factorization of another number is
$ \Rightarrow 378 = 2 \times 3 \times 3 \times 7$
Now we know that the highest common factor of HCF of any two numbers is the common factor of the numbers. All factors of a number are its divisors.
Thus, it gives us the common factors of both the numbers are:
$ \Rightarrow 2 \times 3 \times 3 = 18$
Therefore, the largest number which divides $853$ and $385$ leaving the remainder $7$ is $18$.

Note: We should note that the HCF or highest common factors of two numbers are that number that divides both the numbers without leaving any remainder. Also, we should know that the numbers that have only two factors are called prime numbers. And we know the prime factorization method is a method of expressing numbers as a product of prime factors.