Using Euclid Algorithm to Find HCF of 1651 and 2032

Euclid algorithm is used to find the highest common factor (HCF) of two numbers. Here are the steps to use this algorithm to find the HCF of 1651 and 2032:

1. Divide the larger number by the smaller number.

2. Write down the remainder and divide the smaller number by this remainder.

3. Repeat step 2 until the remainder is zero.

4. The last non-zero remainder is the HCF of the two numbers.

Step-by-Step Solution

Let's use these steps to find the HCF of 1651 and 2032:

1. 2032 ÷ 1651 = 1 remainder 381

2. 1651 ÷ 381 = 4 remainder 67

3. 381 ÷ 67 = 5 remainder 26

4. 67 ÷ 26 = 2 remainder 15

5. 26 ÷ 15 = 1 remainder 11

6. 15 ÷ 11 = 1 remainder 4

7. 11 ÷ 4 = 2 remainder 3

8. 4 ÷ 3 = 1 remainder 1

9. 3 ÷ 1 = 3 remainder 0

The last non-zero remainder is 1. Therefore, the HCF of 1651 and 2032 is 1.

Expressing HCF in the Form 1651m 2032n

Since the HCF is 1, we can express it as 1651m × 2032n, where m and n are any integers.

1 = 1651m × 2032n

We can choose m = 2032 and n = -1651, since:

1651 × (-1651) + 2032 × 2032 = 1

Therefore, the HCF of 1651 and 2032 can be expressed as:

1 = 1651 × 2032 × (-1651) + 2032 × 1651 × 2032

M=5 and n= -4