The LCM of two numbers is 15 times of HCF. The sum of HCF and LCM is 480. If both number are smaller than LCM. Find both the numbers.

Let's denote the HCF as H and the LCM as L.

From the problem, we know that:

1. L = 15H (since the LCM is 15 times the HCF)
2. H + L = 480 (since the sum of the HCF and LCM is 480)

We can substitute equation (1) into equation (2) to solve for H:

H + 15H = 480 16H = 480 H = 480 / 16 H = 30

Substitute H = 30 into equation (1) to find L:

L = 15 * 30 L = 450

So, the HCF is 30 and the LCM is 450.

Now, let's denote the two numbers as a and b.

We know that a * b = H * L (since the product of two numbers is equal to the product of their HCF and LCM)

So, a * b = 30 * 450 a * b = 13500

Since both numbers are smaller than the LCM, they must be factors of 13500 that are less than 450.

The pairs of factors of 13500 are (1, 13500), (2, 6750), (3, 4500), (4, 3375), (5, 2700), (6, 2250), (9, 1500), (10, 1350), (12, 1125), (15, 900), (18, 750), (20, 675), (25, 540), (30, 450), (36, 375), (45, 300), (50, 270), (60, 225), (75, 180), (90, 150), (100, 135), (125, 108), (150, 90), (180, 75), (225, 60), (270, 50), (300, 45), (375, 36), (450, 30), (540, 25), (675, 20), (750, 18), (900, 15), (1125, 12), (1350, 10), (1500, 9), (2250, 6), (2700, 5), (3375, 4), (4500, 3), (6750, 2), (13500, 1).

The pairs that are both less than 450 are (1, 13500), (2, 6750), (3, 4500), (4, 3375), (5, 2700), (6, 2250), (9, 1500), (10, 1350), (12, 1125), (15, 900), (18, 750), (20, 675), (25, 540), (30, 450), (36, 375), (45, 300), (50, 270), (60, 225), (75, 180), (90, 150), (100, 135), (125, 108), (150, 90), (180, 75), (225, 60), (270, 50), (300, 45), (375, 36), (450, 30).

The only pair that has a HCF of 30 (since the HCF is the greatest common factor of the two numbers) is (150, 90).

So, the two numbers are 150 and 90.