Question
The Highest Common Factor (HCF) and Least Common
Multiple (LCM) of two numbers, X and Y, are given as 8 and 80, respectively. Additionally, the difference between the two numbers is 24 (i.e., X - Y = 24). Determine the sum of these two numbers (X + Y).Solution
ATQ,
HCF (X, Y) = 8
And LCM (X, Y) = 80
We know that, HCF (X, Y) × LCM (X, Y) = X × Y
Also, X – Y = 24
Or, Y = X – 24 ……… (I)
ATQ;
X × (X – 24) = 80 × 8
Or, X² – 24X = 640
Or, X² – 24X – 640 = 0
Or, X² – 40X + 16X – 640 = 0
Or, X(X – 40) + 16(X – 40) = 0
Or, (X – 40)(X + 16) = 0
So, X = 40 or X = -16
Since, X cannot Ye negative, we may discard X = -16.
So, X = 40
And, Y = 40 – 24 = 16
Therefore, required sum = (40 + 16) = 56
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