Question

The equation
$a\times b$ = (LCM of a and b $\times$ HCF of a and b) is valid for

• A. integers
• B. fractions
• C. decimals
• D. only natural numbers

Answer 1

Answer: (A), (B), (C)

The correct options are
A

integers

B

fractions

C

decimals

Consider two numbers be 12 and 18.
Highest common factor (HCF) of 12 and 18 is 6.
Lowest common multiple (LCM) of 12 and 18 is 36.
HCF × LCM = 6 × 36 = 216
Also 12 × 18 = 216
Therefore, the product of HCF and LCM of 12 and 18 = product of 12 and 18.

Consider two numbers 0.12 and 0.18
HCF of 0.12 and 0.18 = 0.06
LCM of 0.12 and 0.18 = 0.36
LCM x HCF = 0.06 x 0.36 = 0.0216
Also 0.12 x 0.18 = 0.0216
Therefore, the product of HCF and LCM of 0.12 and 0.18 = product of 0.12 and 0.18.

Consider two fractions $\frac{1}{2}$ and $\frac{1}{3}$
We know that, LCM of the fraction = $\frac{\text{LCM of the numerator}}{\text{HCF of the denominator}}$
$\therefore$ LCM = $\frac{\text{LCM of 1 and 1}}{\text{HCF of 2 and 3}}$

LCM of 1 and 1 = 1 and HCF of 2 and 3 = 1

LCM of $\frac{1}{2}$ and $\frac{1}{3}$ = $\frac{1}{1}=1$

Also, HCF of the fraction = $\frac{\text{HCF of the numerator}}{\text{LCM of the denominator}}$
$\therefore$ LCM = $\frac{\text{HCF of 1 and 1}}{\text{LCM of 2 and 3}}$

HCF of 1 and 1 = 1 and LCM of 2 and 3 = 6

HCF of $\frac{1}{2}$ and $\frac{1}{3}$ = $\frac{1}{6}$
Product of HCF and LCM = $\frac{1}{1}\times \frac{1}{6}=\frac{1}{6}$
And product of the given fractions = $\frac{1}{2}\times \frac{1}{3}=\frac{1\times 1}{2\times 3}=\frac{1}{6}$
Therefore, the product of HCF and LCM of $\frac{1}{2}$ and $\frac{1}{3}$ = product of $\frac{1}{2}$ and $\frac{1}{3}$

$\therefore$ The given equation is applicable for integers, fractions, and decimals.