The L.C.M and H.C.F of two numbers is 585 and 13 respectively. Find the…

To find the difference between two numbers given their Least Common Multiple (LCM) and Highest Common Factor (HCF), we can use the following relationship:

Key Property
If two numbers have an HCF of H, they can be expressed as Ha and Hb, where a and b are coprime (meaning they share no common factors other than 1). The product of the two numbers is equal to the product of their LCM and HCF:
Numbers = (H * a) and (H * b)
LCM = H * a * b

Step-by-Step Calculation
Identify the values:

HCF (H) = 13

LCM = 585

Find the product of the coprime factors (a * b):

LCM = H * a * b

585 = 13 * a * b

a * b = 585 / 13 = 45

Determine coprime pairs for 45:

The pairs of factors for 45 are: (1, 45), (3, 15), and (5, 9).

We need coprime pairs, so we exclude (3, 15) because they share a factor of 3.

Possible coprime pairs are (1, 45) and (5, 9).

Find the numbers and their differences:

Case 1: Pair (1, 45)

Numbers = 13 * 1 = 13 and 13 * 45 = 585

Difference = 585 - 13 = 572

Case 2: Pair (5, 9)

Numbers = 13 * 5 = 65 and 13 * 9 = 117

Difference = 117 - 65 = 52