Use Euclid's division algorithm to find the HCF of 420 and 130 .

Answer

To find the HCF of 420 and 130 using Euclid's division algorithm, we will follow these steps:

### Step 1: Apply Euclid's Division Algorithm
We start with the two numbers, \( a = 420 \) and \( b = 130 \). According to Euclid's division algorithm, we can express \( a \) in terms of \( b \) as follows:

\[ a = bq + r \]

where \( q \) is the quotient and \( r \) is the remainder. We need to find \( q \) and \( r \).

### Step 2: Calculate the first division
Now, we divide 420 by 130:

\[ 420 = 130 \times 3 + r \]

Calculating \( 130 \times 3 = 390 \):

\[ 420 = 390 + r \]

Now, we find \( r \):

\[ r = 420 - 390 = 30 \]

So, we have:

\[ 420 = 130 \times 3 + 30 \]

### Step 3: Apply the algorithm again
Since the remainder \( r \) is not zero, we apply the algorithm again using \( b = 130 \) and \( r = 30 \):

\[ 130 = 30q + r \]

Calculating \( 130 \div 30 \):

\[ 130 = 30 \times 4 + r \]

Calculating \( 30 \times 4 = 120 \):

\[ 130 = 120 + r \]

Now, we find \( r \):

\[ r = 130 - 120 = 10 \]

So, we have:

\[ 130 = 30 \times 4 + 10 \]

### Step 4: Apply the algorithm again
Since the remainder \( r \) is still not zero, we apply the algorithm again using \( b = 30 \) and \( r = 10 \):

\[ 30 = 10q + r \]

Calculating \( 30 \div 10 \):

\[ 30 = 10 \times 3 + 0 \]

Here, the remainder \( r = 0 \).

### Step 5: Conclusion
Since we have reached a remainder of zero, the last non-zero remainder is the HCF. Therefore, the HCF of 420 and 130 is:

\[ \text{HCF} = 10 \]

### Summary
Thus, the HCF of 420 and 130 using Euclid's division algorithm is **10**.

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