Question

Find the largest positive integer that will divide 398, 436 and 542 leaving remainders 7, 11 and 15 respectively.
(a) 15
(b) 16
(c) 17
(d) 18

Answer 1

Hint: We will need to first subtract each given remainder from each given respective number so that number will be perfectly divisible by the largest positive integer we need to find. For example if we divide 10 by 3 then it’s remainder is 1 and if we subtract 1 from 10 i.e. we get 9 which is completely divisible by 3 with 0 remainder. Then we will take HCF of all the three numbers formed to find the largest positive integer that will divide all 3 numbers as HCF itself stands for ‘Highest Common Factor’.

Complete step-by-step answer:
We will first subtract each given remainder from each given respective number so that number will be perfectly divisible by the largest positive integer we need to find, for example if we divide 8 by 5 we will get remainder as 3 and if we subtract 3 from 8 i.e. we get 5 which is completely divisible by 5 with 0 remainder. So, we get
398 – 7 = 391,
436 – 11 = 425,
542 – 15 = 527.
Hence the new numbers we got are 391, 425, 527.
Now we will take HCF of the above three numbers to get the largest positive integer that divides all of three integers,
We can find the HCF by prime factorization method i.e. by breaking each number in it’s prime factor form and finding out which factors are common in all of them, hence
 $ \begin{align}
  & 391=17\times 23 \\
 & 425=5\times 5\times 17 \\
 & 527=17\times 31 \\
\end{align} $
Now we can observe that out of all factors only 17 are common in all of them so the largest common factor that will divide all of them is 17.
Now even we can check that when we divide 398, 436 and 542 by 17 it leaves remainders as 7, 11 and 15 respectively.
So, the correct answer is “Option C”.

Note: You should always remember that whenever questions related to finding a common variable among random numbers are given it is usually solved through LCM or HCF, so try to apply that.
And you can also solve the above question simply by just dividing given numbers by each option and see whether the remainder comes out matches with the question or not. But note that you have to check all of the options as maybe it satisfies the remainder condition but it may not be the largest number satisfying that criteria.