HCF of two numbers is 23 and product of other factors of their LCM is 15.Find the minimum difference between the two numbers .

Sure, here is the step by step solution:

Step 1: The HCF of the two numbers is given as 23. This means that 23 is a factor of both numbers.

Step 2: The product of the other factors of their LCM is given as 15. This means that the other factors of the LCM (apart from 23) are 3 and 5.

Step 3: Since LCM is the product of the highest powers of all the factors involved, the two numbers could be 233 and 235.

Step 4: Therefore, the two numbers are 69 and 115.

Step 5: The difference between the two numbers is 115 - 69 = 46.

So, the minimum difference between the two numbers is 46.

Sure, here is the step by step solution:

Step 1: We know that the Highest Common Factor (HCF) of the two numbers is 23.

Step 2: We also know that the product of the other factors of their Least Common Multiple (LCM) is 15.

Step 3: Since the HCF is a factor of the LCM, we can say that the LCM of the two numbers is 23 * 15 = 345.

Step 4: Let's denote the two numbers as a and b. Since the HCF of a and b is 23, we can write a = 23m and b = 23n, where m and n are integers.

Step 5: Since the LCM of a and b is 345, we can write 345 = 23m * n. Substituting 345/23 = 15 for 23m * n, we get m * n = 15.

Step 6: The pairs of integers (m, n) that satisfy m * n = 15 are (1, 15) and (3, 5).

Step 7: The corresponding pairs of numbers (a, b) are (23, 345) and (69, 115).

Step 8: The differences between the numbers in each pair are 322 and 46.

Step 9: Therefore, the minimum difference between the two numbers is 46.