Express the H.C.F. of 18 and 24 in the form of 18x + 24y.

Answer

To express the H.C.F. of 18 and 24 in the form of \(18x + 24y\), we will follow these steps:

### Step 1: Find the H.C.F. of 18 and 24
To find the H.C.F., we can use the prime factorization method or the division method. Here, we will use the division method.

1. Divide 24 by 18:
\[ 24 \div 18 = 1 \quad \text{(quotient)} \]
\[ \text{Remainder} = 24 - 18 \times 1 = 6 \]

2. Now, divide 18 by the remainder 6:
\[ 18 \div 6 = 3 \quad \text{(quotient)} \]
\[ \text{Remainder} = 18 - 6 \times 3 = 0 \]

Since the remainder is now 0, the last non-zero remainder is the H.C.F. Thus, the H.C.F. of 18 and 24 is 6.

### Step 2: Express the H.C.F. in the form of \(18x + 24y\)
We need to express 6 in the form \(18x + 24y\). We can use the results from our division steps.

From our first division:
\[ 24 = 18 \times 1 + 6 \]
Rearranging gives:
\[ 6 = 24 - 18 \times 1 \]

### Step 3: Rewrite the equation
We can rewrite the equation as:
\[ 6 = 18 \times (-1) + 24 \times 1 \]

### Conclusion
Thus, we have expressed the H.C.F. of 18 and 24 in the form \(18x + 24y\) where \(x = -1\) and \(y = 1\):
\[ \text{H.C.F.} = 18 \times (-1) + 24 \times 1 \]