Question

Find the largest number which divides 445 by 72 and 699 by leaving remainder 456 respectively

Answer 1

If I am right, it must be 572 in place of 72 as I have seen many similar questions before. Please make sure that the question is right again :)

Subtract each of the remainders: 
445 - 4 = 441 
572 - 5 = 567 
699 - 6 = 693 

Now find the greatest common factor of those 3 numbers: 
441 = 3 x 3 x 7 x 7 
572 = 3 x 3 x 3 x 3 x 7 
693 = 3 x 3 x 7 x 11 

The common factors are 3 x 3 x 7 = 63 
gcf(441,567,693) = 63 

Double-check: 
445 / 63 = 7 remainder 4 
572 / 63 = 9 remainder 5 
699 / 63 = 11 remainder 6 

Answer: 
63 is the largest divisor that will give the desired remainders.

There is an alternate method too:

The largest number that will divide 699, 572 and 445 leaving remainders 6, 5 and 4 respectively is the HCF of the numbers (699 – 6), (572 – 5) and (445 – 4) i.e. 693, 567 and 441.

HCF of 693, 567 and 441:
HCF of 693 and 567:
693 = 567 x 1 + 126
567 = 126 x 4 + 63
126 = 63 x 2 + 0
HCF of 693 and 567 = 63

Similarly, HCF of 63 and 441= 63
Thus, HCF of 693, 567 and 441 = 63

∴ The largest number that will divide 699, 572 and 445 leaving remainders 6, 5 and 4 respectively is 63.

Or, 

Given numbers : 445,572 and 699. 

On dividing these given numbers by the greatest number, we get remainder as 4 ,5 and 6 respectively.

From the given numbers, we now subtract their respective remainders:

445- 4 = 441

 572 - 5 = 567  

And 699 - 6 = 693.

Now, we will take the H.C.F of  441, 567 and 693 which is 63.

Therefore, 63 is the greatest number required. 

Hope This Helps :)