**LCM Model 1:**

If $r$ is the remainder in each case when $N$ is divided by $x, y, z$ then the general format of the number is $$N = K \times \left( {LCM(x,y,z)} \right) + r$$ Here $K$ is a natural number.

**LCM Model 2:**

If ${x_1},{y_1},{z_1}$ are the remainders when $N$ is divided by $x, y, z$ and $x - {x_1} = y - {y_1} = z - {z_1} = a$ then the general format of the number is given by $$N = K \times \left( {LCM(x,y,z)} \right) - a$$.

**HCF Model 1:**

$a, b, c$ are the remainders in each case when $A, B, C$ are divided by $N$ then $$N = HCF (A-a, B-b, C-c)$$

**HCF Model 2:**

When $A, B, C$ are divided by $N$ then the remainder is same in each case but remainder is not given then \[N{\text{ }} = {\text{ }}HCF{\text{ }}of{\text{ }}any{\text{ }}two{\text{ }}of{\text{ }}\left( {A - B,{\text{ }}B - C,{\text{ }}C - A} \right)\]

**Formula 1:**

The product of two numbers is equal to the product of LCM and HCF of the two given numbers.

Let $A, B$ are two numbers and $h$ is the HCF of the two numbers.

\[{\text{A}} \times {\text{B = LCM (A, B) }} \times {\text{ HCF (A, B)}}\]

\[{\text{A}} \times {\text{B = LCM (A, B) }} \times {\text{ HCF (A, B)}}\]

**Formula 2:**

LCM of the Fractions = \(\dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}\)

**Formula 3:**

HCF of the Fractions = \(\dfrac{\text{HCF of numerators}}{\text{LCM of denominators}}\)

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