NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.1

Understanding the Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic is the foundation of Real Numbers Ex 1.1. It states that: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. For example, the composite number 30 can be written as $2 \times 3 \times 5$. No matter how you order these prime numbers, the prime factors of 30 will always consist of exactly one 2, one 3, and one 5. This theorem helps us uniquely identify any integer and serves as a powerful tool to find LCM and HCF of given numbers efficiently.

Step-by-Step Method: Prime Factorization & Verification

1. Step 1: Express Numbers as Prime Factors — Divide each number by successive prime numbers (2, 3, 5, 7, etc.) until you get 1. Express the number as a product of its prime factors in exponential form.
2. Step 2: Find the HCF — Identify the common prime factors. The HCF is the product of the lowest power of each common prime factor involved in the numbers.
3. Step 3: Find the LCM — List all the prime factors present in any of the numbers. The LCM is the product of the highest power of each prime factor involved in the numbers.
4. Step 4: Verify the Relationship — For any two positive integers $a$ and $b$, check if $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$. Note that this rule only applies to a pair of two numbers, not three.

Step-by-Step Worked Examples

• Example 1: Express 140 as a product of prime factors. Divide 140 by 2: $140 \div 2 = 70$ Divide 70 by 2: $70 \div 2 = 35$ Divide 35 by 5: $35 \div 5 = 7$ 7 is a prime number: $7 \div 7 = 1$ * Therefore, prime factorization of $140 = 2 \times 2 \times 5 \times 7 = 2^2 \times 5 \times 7$.
• Example 2: Find HCF and LCM of 26 and 91, and verify that $\text{LCM} \times \text{HCF} = \text{product of the two numbers}$. Prime factorization of $26 = 2 \times 13$ Prime factorization of $91 = 7 \times 13$ Finding HCF: The common prime factor is 13, and its lowest power is 1. Thus, $\text{HCF}(26, 91) = 13$. Finding LCM: The prime factors involved are 2, 7, and 13. Their highest powers are 1. Thus, $\text{LCM}(26, 91) = 2 \times 7 \times 13 = 182$. Verification: $\text{HCF} \times \text{LCM} = 13 \times 182 = 2366$ $\text{Product of numbers} = 26 \times 91 = 2366$ Since $2366 = 2366$, the formula is verified!

Crucial Board Exam Tips & Pitfalls

1. The Three-Number Trap: Remember that $\text{HCF}(a, b, c) \times \text{LCM}(a, b, c) \neq a \times b \times c$. This formula only works for exactly two numbers. Do not attempt to use it for three-number problems.
2. Expressing Powers correctly: When calculating HCF, always select the lowest power of the common base. For LCM, select the highest power of every base. Misidentifying the power is the most common reason for calculation errors.
3. Verify by multiplication: Always check your calculations. A quick way to test if your HCF is correct is to verify that it divides both numbers perfectly.

Practice Questions with Solutions

• Q: Express the composite number 156 as a product of its prime factors. A: Step 1: Perform division by prime numbers: - $156 \div 2 = 78$ - $78 \div 2 = 39$ - $39 \div 3 = 13$ - 13 is a prime number, so $13 \div 13 = 1$ Step 2: Collect all the prime factors: $156 = 2 \times 2 \times 3 \times 13$ Step 3: Express in exponential form: $156 = 2^2 \times 3^1 \times 13^1$ Final answer: $156 = 2^2 \times 3 \times 13$
• Q: Find the LCM and HCF of 510 and 92, and verify that LCM $\times$ HCF = product of the two numbers. A: Step 1: Write down prime factorization: - $510 = 2 \times 3 \times 5 \times 17$ - $92 = 2^2 \times 23$ Step 2: Calculate HCF (lowest power of common factor): - Common factor is 2. Lowest power is $2^1$. - $\text{HCF} = 2$ Step 3: Calculate LCM (highest power of all factors present): - $\text{LCM} = 2^2 \times 3 \times 5 \times 17 \times 23 = 4 \times 3 \times 5 \times 17 \times 23 = 23460$ Step 4: Verify: - $\text{LCM} \times \text{HCF} = 23460 \times 2 = 46920$ - $\text{Product of numbers} = 510 \times 92 = 46920$ - Since $46920 = 46920$, the relationship holds true. Final answer: HCF = 2, LCM = 23460, Verified.
• Q: Find the HCF and LCM of 12, 15 and 21 using the prime factorisation method. A: Step 1: Find the prime factorization of each number: - $12 = 2^2 \times 3$ - $15 = 3 \times 5$ - $21 = 3 \times 7$ Step 2: Find HCF by taking the lowest power of common factors. The only common factor is 3. - $\text{HCF} = 3^1 = 3$ Step 3: Find LCM by taking the highest power of all prime factors present ($2^2, 3^1, 5^1, 7^1$): - $\text{LCM} = 2^2 \times 3 \times 5 \times 7 = 4 \times 3 \times 5 \times 7 = 420$ Final answer: HCF = 3, LCM = 420
• Q: Given that HCF(306, 657) = 9, find the LCM of 306 and 657. A: Step 1: Use the relationship formula for two positive integers: $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$ Step 2: Substitute the known values ($a = 306$, $b = 657$, $\text{HCF} = 9$): $9 \times \text{LCM}(306, 657) = 306 \times 657$ Step 3: Solve for LCM: $\text{LCM}(306, 657) = \frac{306 \times 657}{9}$ Step 4: Simplify the expression: - Divide 306 by 9: $306 \div 9 = 34$ - Calculate $34 \times 657 = 22338$ Final answer: LCM(306, 657) = 22338

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