Find the HCF of 36 and 45 by short division method.

To find the Highest Common Factor (HCF) of 36 and 45 using the short division method, we can follow a systematic approach. This method involves dividing the numbers by their common prime factors until we can no longer divide. Let’s break it down step by step.

Step-by-Step Process

1. Identify the Numbers

We are working with the numbers 36 and 45. The goal is to find the HCF, which is the largest number that divides both without leaving a remainder.

2. Start with the Short Division

We begin by dividing both numbers by the smallest prime number, which is 2. However, since 36 is even and 45 is odd, we can only divide 36 by 2:

• 36 ÷ 2 = 18
• 45 cannot be divided by 2, so we move to the next prime number, which is 3.

3. Continue Dividing by Prime Numbers

Now, we divide both numbers by 3:

• 18 ÷ 3 = 6
• 45 ÷ 3 = 15

Next, we can divide again by 3:

• 6 ÷ 3 = 2
• 15 ÷ 3 = 5

4. Final Division

At this point, we have:

• 2 (which is a prime number)
• 5 (which is also a prime number)

Since 2 and 5 cannot be divided further by any common prime factor, we stop here.

Collecting the Common Factors

Now, let’s summarize the prime factors we used:

• From 36: 2 and 3 (twice, since we divided by 3 two times)
• From 45: 3 (once)

The common prime factor is 3. To find the HCF, we multiply the common factors:

5. Calculate the HCF

Since the only common prime factor is 3, the HCF of 36 and 45 is:

HCF = 3

Conclusion

In summary, using the short division method, we found that the HCF of 36 and 45 is 3. This method is efficient for finding the HCF, especially when dealing with larger numbers or multiple numbers at once. If you have any further questions or need clarification on any of the steps, feel free to ask!