Find the smallest pair of 4-digit numbers such that the difference between them is 303 and their HCF is 101. Show your steps.

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Identify Numbers: Since the HCF of the two numbers is 101, both numbers must be multiples of 101. Let's denote the smaller number as 101a and the larger number as 101b, where a and b are integers.Set Up Equation: The difference between the two numbers is given as 303, which is also a multiple of 101 (since 303 = 101 * 3). We can set up the equation 101b - 101a = 303 to find the relationship between a and b.Solve for Relationship: Dividing both sides of the equation by 101, we get b - a = 3. This means that the two integers a and b must be 3 units apart.Find Smallest a: Since we are looking for the smallest 4-digit numbers, we want to find the smallest possible value for a such that 101a is a 4-digit number. The smallest 4-digit number is 1000, so we need to find the smallest integer a such that 101a ≥ 1000.Calculate Minimum a: Dividing 1000 by 101, we get approximately 9.9. Since a must be an integer, the smallest possible value for a that makes 101a a 4-digit number is 10.Determine Values of a and b: Now that we have a = 10, we can find b by adding 3 to a, giving us b = 10 + 3 = 13.Calculate Numbers: The two numbers are therefore 101a = 101 * 10 = 1010 and 101b = 101 * 13 = 1313.Verify Solution: We check that the difference between the two numbers is 1313 - 1010 = 303, which matches the condition given in the problem.

Find the smallest pair of $4$ -digit numbers such that the difference between them is $303$ and their HCF is $101$ . Show your steps.

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Find the smallest pair of $4$ -digit numbers such that the difference between them is $303$ and their HCF is $101$ . Show your steps.