What is the greatest number that, when divided by 460, 491, and 553, leaves 26 as a remainder in each case?

We have to find the greatest number, which, when divided by 460, 491, and 553, leaves 26 as the remainder in each case.

If the number is assumed to be x, then 460 on division by x leaves remainder 26.

So 460 - 26 = 434 is divisible by x. 

Similarly, 

491 - 26 = 465 is divisible by x

and, 553 - 26 = 527 is also divisible by x

The greatest number that will divide all three numbers is the HCF of the numbers. 

Taking HCF (434, 465, 527) = 31 

So, the greatest number that, when divides 460, 491 and 553, leaves 26 as a remainder in each case, is 31.