A boy divided the numbers 7654, 8506 and 9997 by a certain largest number and he gets same remainder in each case. What is the common remainder?

• Ans: 199
  
  If the remainder is same in each case and remainder is not given, HCF of the differences of the numbers is the required largest number
  9997 - 7654 = 2343
  9997 - 8506 = 1491
  8506 - 7654 = 852
  Hence, the greatest number which divides 7654, 8506 and 9997 and leaves same remainder
  
  = HCF of 2343, 1491, 852
  
  = 213
  Now we need to find out the common remainder.
  Take any of the given numbers from 7654, 8506 and 9997, say 7654
  
  7654 ÷ 213 = 35, remainder = 199
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