Express the HCF of numbers 72 and 124 as a linear combination of 72 and 124.

Answer

To express the HCF of 72 and 124 as a linear combination of these two numbers, we will follow these steps:

### Step 1: Find the HCF of 72 and 124
We will use the Euclidean algorithm to find the HCF.

1. Divide 124 by 72:
\[ 124 \div 72 = 1 \quad \text{(remainder 52)} \]
So, \( 124 = 72 \times 1 + 52 \).

2. Now, divide 72 by 52:
\[ 72 \div 52 = 1 \quad \text{(remainder 20)} \]
So, \( 72 = 52 \times 1 + 20 \).

3. Next, divide 52 by 20:
\[ 52 \div 20 = 2 \quad \text{(remainder 12)} \]
So, \( 52 = 20 \times 2 + 12 \).

4. Now, divide 20 by 12:
\[ 20 \div 12 = 1 \quad \text{(remainder 8)} \]
So, \( 20 = 12 \times 1 + 8 \).

5. Next, divide 12 by 8:
\[ 12 \div 8 = 1 \quad \text{(remainder 4)} \]
So, \( 12 = 8 \times 1 + 4 \).

6. Finally, divide 8 by 4:
\[ 8 \div 4 = 2 \quad \text{(remainder 0)} \]
So, \( 8 = 4 \times 2 + 0 \).

Since the remainder is now 0, the HCF of 72 and 124 is 4.

### Step 2: Express the HCF as a Linear Combination
We need to express 4 as a linear combination of 72 and 124, i.e., find integers \( a \) and \( b \) such that:
\[ 4 = 72a + 124b \]

We can backtrack from our previous divisions to find \( a \) and \( b \).

1. From \( 12 = 8 \times 1 + 4 \), we can express 4:
\[ 4 = 12 - 8 \times 1 \]

2. Substitute \( 8 \) from \( 20 = 12 \times 1 + 8 \):
\[ 8 = 20 - 12 \times 1 \implies 4 = 12 - (20 - 12) = 2 \times 12 - 20 \]

3. Substitute \( 12 \) from \( 52 = 20 \times 2 + 12 \):
\[ 12 = 52 - 20 \times 2 \implies 4 = 2(52 - 20 \times 2) - 20 = 2 \times 52 - 4 \times 20 \]

4. Substitute \( 20 \) from \( 72 = 52 \times 1 + 20 \):
\[ 20 = 72 - 52 \implies 4 = 2 \times 52 - 4(72 - 52) = 2 \times 52 - 4 \times 72 + 4 \times 52 = 6 \times 52 - 4 \times 72 \]

5. Substitute \( 52 \) from \( 124 = 72 \times 1 + 52 \):
\[ 52 = 124 - 72 \implies 4 = 6(124 - 72) - 4 \times 72 = 6 \times 124 - 6 \times 72 - 4 \times 72 = 6 \times 124 - 10 \times 72 \]

Thus, we have:
\[ 4 = 6 \times 124 - 10 \times 72 \]

### Final Answer
The HCF of 72 and 124 expressed as a linear combination is:
\[ 4 = -10 \times 72 + 6 \times 124 \]