Question

Find the largest positive integer that will divide $398,436$ and $542$ leaving remainders $7,11$ and $15$ respectively.

Answer 1

It is given that on dividing $398$ by the required number, there is a remainder of $7$.

This means that $398-7=391$ is exactly divisible by the required number. In other words, the required number is a factor of $391$.

Similarly, required positive integer is a factor of $436-11=425$ and $542-15=527$.
Clearly, required number is the $HCF$ of $391,425$ and $527$.

Using the prime factorizations of $391,425$ and $527$ are as follows:

$391=17\times 23,425=5^2\times 17$ and $527=17\times 31$

$\therefore HCF$ of $391,425$ and $527$ is $17$

Hence, required number $=17$