Express the H.C.F. of 30 and 36 in the form of 30x + 36y.

Answer

To express the H.C.F. of 30 and 36 in the form of \(30x + 36y\), we will follow these steps:

### Step 1: Find the H.C.F. of 30 and 36
To find the H.C.F. (Highest Common Factor) of 30 and 36, we can use the prime factorization method.

- **Prime factorization of 30**:
\[ 30 = 2 \times 3 \times 5 \]
- **Prime factorization of 36**:
\[ 36 = 2^2 \times 3^2 \]

The common prime factors are \(2\) and \(3\). The lowest powers of these common factors are:
- For \(2\): \(2^1\)
- For \(3\): \(3^1\)

Thus, the H.C.F. is:
\[ H.C.F. = 2^1 \times 3^1 = 6 \]

### Step 2: Express H.C.F. in the form \(30x + 36y\)
We need to express \(6\) in the form \(30x + 36y\).

From the equation:
\[ 6 = 30x + 36y \]

We can rearrange it to find integer values for \(x\) and \(y\).

### Step 3: Find integer solutions for \(x\) and \(y\)
We can express \(6\) in terms of \(30\) and \(36\).

Let’s start by manipulating the equation:
\[ 6 = 36 \cdot 1 - 30 \cdot 1 \]

This gives us:
\[ 6 = 30(-1) + 36(1) \]

From this equation, we can see that:
- \(x = -1\)
- \(y = 1\)

### Final Expression
Thus, we can express the H.C.F. of 30 and 36 in the required form:
\[ 6 = 30(-1) + 36(1) \]

### Summary
The H.C.F. of 30 and 36 is \(6\), and it can be expressed as:
\[ 6 = 30(-1) + 36(1) \]