The numbers 11284 and 7655, when divided by a certain number of three digits, leave the same remainder. Find that number of three digits.a)161b)171c)181d)191Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Problem:
The numbers 11284 and 7655, when divided by a certain number of three digits, leave the same remainder. Find that number of three digits.

Solution:
To find the three-digit number that divides both 11284 and 7655 with the same remainder, we can use the concept of remainders and modular arithmetic.

Step 1: Finding the Remainders
Let's assume the three-digit number we are looking for is 'x'. When we divide 11284 by 'x', the remainder will be 'r1'. Similarly, when we divide 7655 by 'x', the remainder will be 'r2'.

Step 2: Setting up Equations
We can set up two equations based on the given information:
11284 ≡ r1 (mod x) ...(1)
7655 ≡ r2 (mod x) ...(2)

Step 3: Simplifying the Equations
Since the remainders are the same for both numbers, we can express them as a single variable:
r1 = r2 = r

Now, we can rewrite equations (1) and (2) as:
11284 ≡ r (mod x) ...(3)
7655 ≡ r (mod x) ...(4)

Step 4: Using the Principle of Congruence
From equations (3) and (4), we can conclude that 11284 and 7655 are congruent to each other modulo x. This can be expressed as:
11284 ≡ 7655 (mod x) ...(5)

Step 5: Finding the Common Divisor
To find the common divisor, we need to find the greatest common divisor (GCD) of 11284 and 7655. If the GCD is a three-digit number, then it satisfies the given condition.

Using a calculator or a mathematical tool, we find that the GCD of 11284 and 7655 is 191.

Therefore, the three-digit number that divides both 11284 and 7655 with the same remainder is 191.

Conclusion:
The correct answer is option (D) 191.

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