find the greatest number which on dividing 1657 and 2037 leaves remainder 6 and 5 respectively.
In this question why do we have to subtract the remainders and what will happen if we dont subtract them? 
How will it affect the ans.
Plz explain briefly
Thank you!

 Solution :

We have the two numbers which are: 1657 and 2037.
When we divide these numbers by the required number, we have the remainders of 6 and 5 respectively.
So, let us first just remove these remainders to get a clean and perfect division.
Now, we will have: 1657 – 6 = 1651
And 2037 – 5 = 2032.
Now, we have created two new numbers 1651 and 2032 which are perfectly divisible by the number we are required to find.
If two numbers are divisible by some number that means that number is a factor of both the numbers.
The number is given to be greatest.
Hence, we just need to find HCF.
So, let us first do the prime factorization of 1651=13×127 and  2032=2×2×2×2×127 
We see that they only have 127 common in their prime factorization.
Hence, HCF(1651, 2032) = 127.
Hence, the greatest number which divides 1657 and 2037 leaves a remainder of 6 and 5 respectively is 127.
We have to subtract remainder because above it was mentioned that the number divides the given number and leaves a remainder that means remainder is subtracted to get the number which is divisible by the largest number.
and if we don't subtract remainders we can't find the largest number which when divided by both the number leaves remainders.

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