The product of 2 numbers is 6750 and their LCM is 450. If difference between the numbers is equal to their HCF, then find the smaller number

Sure, let's solve this step by step.

Step 1: We know that the product of two numbers is equal to the product of their LCM and HCF. So, we can write it as:

LCM * HCF = Product of numbers
450 * HCF = 6750

Step 2: Solve the above equation to find the HCF:

HCF = 6750 / 450
HCF = 15

Step 3: We know that the difference between the numbers is equal to their HCF. So, the difference between the numbers is 15.

Step 4: Let's denote the numbers as x and y. We can write the following system of equations according to the problem:

x * y = 6750 (since the product of the numbers is 6750)
x - y = 15 (since the difference between the numbers is equal to their HCF)

Step 5: Solve this system of equations. You can use substitution or elimination method. Let's use substitution:

From the second equation, we can express x as x = y + 15. Substitute x in the first equation:

(y + 15) * y = 6750
y^2 + 15y - 6750 = 0

Step 6: Solve this quadratic equation. You can use the quadratic formula:

y = [-b ± sqrt(b^2 - 4ac)] / (2a)

In our case, a = 1, b = 15, c = -6750. So,

y = [-15 ± sqrt((15)^2 - 4*1*(-6750))] / (2*1)
y = [-15 ± sqrt(225 + 27000)] / 2
y = [-15 ± sqrt(27225)] / 2
y = [-15 ± 165] / 2

Step 7: We get two solutions: y = 75 and y = -90. Since y cannot be negative, we discard -90. So, y = 75.

Step 8: Substitute y = 75 in the equation x = y + 15 to find x:

x = 75 + 15
x = 90

So, the numbers are 75 and 90. The smaller number is 75.