Playing with Numbers Ex 3.5 Class 6 NCERT Maths: Prime Factorization, HCF, and LCM

Understanding Prime Factorization

Before we jump into calculations, let's refresh our memory on some key terms. A prime number is a whole number greater than 1 that has only two factors: 1 and itself (examples: 2, 3, 5, 7, 11). A composite number is a whole number greater than 1 that has more than two factors (examples: 4, 6, 8, 9, 10). Now, prime factorization is the process of expressing a composite number as a product of its prime factors. Think of it like disassembling a toy into its smallest, indivisible parts. Every composite number has a unique prime factorization. This unique property, known as the Fundamental Theorem of Arithmetic, makes prime factorization a powerful tool for understanding numbers and their relationships. For instance, knowing the prime factors of numbers helps us find common factors and multiples easily, which we'll explore next. For example, the number 12 can be written as $2 \times 2 \times 3$. Here, 2 and 3 are prime numbers, and they are the prime factors of 12.

Step-by-Step: Finding Prime Factorization Using Division Method

1. Step 1: Start with the Smallest Prime Number — Begin by dividing the given number by the smallest prime number (which is 2) if it's divisible. If not, try the next smallest prime number (3), then 5, and so on.
2. Step 2: Continue Dividing — Keep dividing the quotient by the same prime number as long as it's divisible. If it's no longer divisible by that prime number, move to the next smallest prime number.
3. Step 3: Repeat Until Quotient is 1 — Continue this process until the quotient becomes 1. The prime factors are all the divisors you used in the process.
4. Example: Find the prime factorization of 72 — 1. Divide 72 by 2: $72 \div 2 = 36$ 2. Divide 36 by 2: $36 \div 2 = 18$ 3. Divide 18 by 2: $18 \div 2 = 9$ 4. Now, 9 is not divisible by 2, so move to the next prime, 3. Divide 9 by 3: $9 \div 3 = 3$ 5. Divide 3 by 3: $3 \div 3 = 1$ So, the prime factorization of 72 is $2 \times 2 \times 2 \times 3 \times 3$ or $2^3 \times 3^2$.

Using Prime Factorization for HCF and LCM

• Example 1: Find the HCF of 24 and 36 1. Prime factorization of 24: $2 \times 2 \times 2 \times 3$ 2. Prime factorization of 36: $2 \times 2 \times 3 \times 3$ 3. Identify common prime factors: Both have $2 \times 2 \times 3$ in common. 4. Multiply the common prime factors: $2 \times 2 \times 3 = 12$. Therefore, the HCF of 24 and 36 is 12.
• Example 2: Find the LCM of 12 and 18 1. Prime factorization of 12: $2 \times 2 \times 3$ 2. Prime factorization of 18: $2 \times 3 \times 3$ 3. List all prime factors from both numbers, taking the highest power of each: $2^2$ (from 12) and $3^2$ (from 18). 4. Multiply these highest powers: $2^2 \times 3^2 = (2 \times 2) \times (3 \times 3) = 4 \times 9 = 36$. Therefore, the LCM of 12 and 18 is 36.

YoLearn's Exam Tip: Double-Check Your Factors!

When performing prime factorization, it's very common to accidentally use a composite number as a factor (e.g., dividing by 4 instead of $2 \times 2$). Always ensure that every factor you use in your division method is a prime number. Also, don't forget to check your work by multiplying all your prime factors together; the result should be the original number. For HCF and LCM, make sure you correctly identify the common factors for HCF and the highest powers of all distinct factors for LCM. A small mistake in identifying factors can lead to an incorrect final answer.

Practice Questions with Solutions

• Q: Find the prime factorization of 90. A: Step 1: Start dividing 90 by the smallest prime number, 2. $90 \div 2 = 45$. Step 2: 45 is not divisible by 2. Try the next prime, 3. $45 \div 3 = 15$. Step 3: 15 is divisible by 3. $15 \div 3 = 5$. Step 4: 5 is a prime number. $5 \div 5 = 1$. Final answer: The prime factorization of 90 is $2 \times 3 \times 3 \times 5$ or $2 \times 3^2 \times 5$.
• Q: Find the HCF of 48 and 72 using prime factorization. A: Step 1: Prime factorize 48: $48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3$. Step 2: Prime factorize 72: $72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$. Step 3: Identify common prime factors with the lowest power: $2^3$ and $3^1$. Step 4: Multiply these common factors: $2^3 \times 3 = 8 \times 3 = 24$. Final answer: The HCF of 48 and 72 is 24.
• Q: Find the LCM of 15 and 25 using prime factorization. A: Step 1: Prime factorize 15: $15 = 3 \times 5$. Step 2: Prime factorize 25: $25 = 5 \times 5 = 5^2$. Step 3: Identify all distinct prime factors with their highest powers: $3^1$ (from 15) and $5^2$ (from 25). Step 4: Multiply these highest powers: $3 \times 5^2 = 3 \times 25 = 75$. Final answer: The LCM of 15 and 25 is 75.
• Q: Can a prime number have more than two factors? Explain. A: Step 1: Recall the definition of a prime number. Step 2: A prime number is defined as a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. Step 3: If a number had more than two factors, it would fit the definition of a composite number, not a prime number. Final answer: No, a prime number cannot have more than two factors. By definition, a prime number has exactly two factors: 1 and the number itself.

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