Find the HCF Of 165 AND 220 USING COMMON DIVISION METHOD
The correct answer and explanation is:
Sure! Let’s find the HCF (Highest Common Factor) of 165 and 220 using the Common Division Method.
Step-by-step solution:
Step 1: Find the common prime factors by dividing both numbers by common divisors.
Start with the smallest prime number, 2, and check if both numbers are divisible.
- 165 ÷ 2 = 82.5 (not divisible)
- 220 ÷ 2 = 110 (divisible)
Since 165 is not divisible by 2, move to the next prime number: 3.
- 165 ÷ 3 = 55 (divisible)
- 220 ÷ 3 ≈ 73.33 (not divisible)
No common factor 3.
Try 5:
- 165 ÷ 5 = 33 (divisible)
- 220 ÷ 5 = 44 (divisible)
Both divisible by 5, so divide both by 5:
- 165 ÷ 5 = 33
- 220 ÷ 5 = 44
Current common factor = 5
Step 2: Now find HCF of 33 and 44
Try 2:
- 33 ÷ 2 = 16.5 (not divisible)
- 44 ÷ 2 = 22 (divisible)
No common factor 2.
Try 3:
- 33 ÷ 3 = 11 (divisible)
- 44 ÷ 3 ≈ 14.67 (not divisible)
No common factor 3.
Try 5:
- 33 ÷ 5 = 6.6 (not divisible)
- 44 ÷ 5 = 8.8 (not divisible)
No common factor 5.
Try 7:
- 33 ÷ 7 ≈ 4.7 (no)
- 44 ÷ 7 ≈ 6.29 (no)
Try 11:
- 33 ÷ 11 = 3 (divisible)
- 44 ÷ 11 = 4 (divisible)
Both divisible by 11, so divide both by 11:
- 33 ÷ 11 = 3
- 44 ÷ 11 = 4
Add 11 to the common factors.
Step 3: Now find HCF of 3 and 4
- 3 and 4 have no common divisors except 1.
Step 4: Multiply all common factors found
Common factors: 5 × 11 = 55
Final answer:
55\boxed{55}
The Highest Common Factor (HCF) of 165 and 220 is 55.
Explanation of the Common Division Method:
The Common Division Method is a way to find the HCF of two or more numbers by dividing them simultaneously by common prime factors, starting from the smallest prime number. You continue dividing the numbers until no further common division is possible. The product of all the common divisors used is the HCF.
This method works because the HCF is the largest number that divides both original numbers exactly without leaving a remainder. By breaking down both numbers into their prime factors and finding the common ones, we isolate the factors that contribute to their greatest shared divisor.
In our example:
- Both 165 and 220 share 5 and 11 as factors.
- After factoring out 5 and 11, what remains (3 and 4) share no common factor except 1.
- Multiplying these common factors gives the HCF, 55.
This method is especially useful for larger numbers or when a systematic approach is needed to find the greatest factor common to both numbers.