Answer

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**Hint:**First we have to define what the terms we need to solve the problem are.

Since we need the given numbers to find the remainder of the same, we need to know about the division first which is if we divide a number by one another number which yields quotient and remainder.

Also, division is the inverse form of the multiplication operator.

**Complete step by step answer:**

First of all, we need to find a three-digit number X such that will given as the answer for the given problem so that by division operation $ \dfrac{{11284}}{X} $ and $ \dfrac{{7655}}{X} $ will need leave us the same remainder \[R\]

Or else by division algorithm; take a quotient $ aX + R = 11284 $ and for the second number $ bX + R = 7655 $ where a and b are natural numbers

Now subtract the equation one minus two we get; $ aX + R = 11284 - (bX + R = 7655) $

So that we get a different constant multiplication terms so we will find the remainder for it which is $ (a - b)X = 3629 $ now we will check the division terms for this constant (common terms)

Now we find the X we have to able to find the three-digit factors, we will calculate that with a prime number that is since two does not divide the term, as continue to all prime number on approaching to $ 19 $ will be divided that means $ 3629 = 19 \times 191 $ or $ \dfrac{{3629}}{{19}} = 191 $ same resultant;

Thus, the only option which satisfies is $ B) $ $ 191 $ when divided by the numbers $ 11284 $ and $ 7655 $ , number of three digits, leave the same remainder.

**So, the correct answer is “Option B”.**

**Note:**Since in the given question they where only asking about the three-digit number and hence we get that \[191\] but also \[19\] has when divided by the numbers $ 11284 $ and $ 7655 $ , number of two digits, leave the same remainder.

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