345 find the HCF and LCM by division method and prime factorization method
The Correct Answer and Explanation is:
To find the HCF (Highest Common Factor) and LCM (Least Common Multiple) of 345 using both the Division Method and the Prime Factorization Method, let’s break it down step by step.
✅ 1. Prime Factorization Method
Step 1: Find the prime factors of 345
- 345 ÷ 3 = 115
- 115 ÷ 5 = 23
- 23 is a prime number
👉 So, the prime factors of 345 are:
345 = 3 × 5 × 23
Since there is only one number (345), the HCF = 345 and LCM = 345 by definition, as both are the same when dealing with a single number.
✅ 2. Division Method (only applies when finding HCF/LCM between two or more numbers)
Since 345 is a single number, the division method is not applicable unless it’s being compared with another number.
📌 Explanation
The concepts of HCF (Highest Common Factor) and LCM (Least Common Multiple) are fundamental in arithmetic and number theory. When we are given only one number, such as 345, both its HCF and LCM are the number itself. This is because:
- The HCF of a single number is the greatest number that divides it without leaving a remainder. Obviously, the only number that divides 345 fully is 345 itself, and smaller divisors do not qualify as “common” factors in this context since no second number is present for comparison.
- The LCM, on the other hand, is the smallest number into which the given number divides without a remainder. Again, 345 divides itself perfectly, and there is no smaller multiple of it than itself.
To explore the structure of the number, we use prime factorization, which breaks the number down into a product of prime numbers. For 345:
- Divide by 3 → 345 ÷ 3 = 115
- Then divide 115 by 5 → 115 ÷ 5 = 23
- 23 is prime
So, 345 = 3 × 5 × 23. This factorization is useful for comparing with other numbers if needed later.
In contrast, the division method (repeated division by common divisors) is used only when finding the HCF of two or more numbers. Since we have only one number here, the method is not applicable.
Thus, for a single number like 345:
✅ HCF = 345
✅ LCM = 345
