VerifiedHint: Here we go through by finding the HCF of the number after subtracting their remainder because we know that The largest or greatest factor common to any two or more given natural numbers is termed as HCF of given numbers. Also known as GCD (Greatest Common Divisor).
Complete answer:
Here in this question for finding the numbers that will divide $398$, $436$, and $542$ leaving remainders $7$, $11$, and $15$ respectively we have to first subtract the remainder of the following. By this step, we find the highest common factor of the numbers.
And then the required number is the HCF of the following numbers that are formed when the remainder are subtracted from them.
Clearly, the required number is the HCF of the numbers $398−7=391,436−11=425$ and, $542−15=527$ .
We will find the HCF of $391$, $425$ and $527$ by prime factorization method.
$391 = 17 \times 23$
$425 = {5^2} \times 17$
$527 = 17 \times 31$
Hence, HCF of $391$, $4250$, and $527$ is $17$ because the greatest common factor from all the numbers is 17 only.
So we can say that the largest number that will divide $398$, $436$ and $542$ leaving remainders $7$, $11$ and $15$ respectively is $17$.
Note: Whenever we face such a type of question the key concept for solving this question is whenever in the question it is asking about the largest number it divides. You should always think about the highest common factor i.e. HCF. we have to subtract remainder because you have to find a factor that means it should be perfectly divisible so to make divisible we subtract remainder. because remainder is the extra number so on subtracting remainder it becomes divisible.
