Discussion
What is the largest number that divides 190, 251 and 373 leaving the same remainder in each case?
Answer: D
Correct answer: 61
When a divisor leaves the same remainder for several numbers, it must divide exactly the differences between them - because the equal remainders cancel on subtraction. The differences are \[ 251 - 190 = 61, \qquad 373 - 251 = 122, \qquad 373 - 190 = 183. \] The required number is the HCF of these: 61 divides 122 and 183 exactly, so \[ \text{HCF}(61, 122, 183) = 61. \] Checking, each of 190, 251 and 373 leaves a remainder of 7 on division by 61. The technique is general and worth remembering in both directions. Where the remainders are equal but unknown, take the HCF of the differences. Where the remainders are given and different, subtract each remainder from its number first and then take the HCF of the results. Taking the HCF of the original numbers rather than the differences is the usual error.
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