We divide 140 successively by the smallest possible prime numbers: 140 ÷ 2 = 70, 70 ÷ 2 = 35, 35 ÷ 5 = 7, and 7 ÷ 7 = 1. Thus, the prime factorization of 140 is 2² × 5 × 7.

Dividing 156 by prime numbers gives 156 ÷ 2 = 78, 78 ÷ 2 = 39, 39 ÷ 3 = 13, and 13 ÷ 13 = 1. Therefore, 156 can be expressed as 2² × 3 × 13.

Since the sum of digits is 3 + 8 + 2 + 5 = 18, it is divisible by 3: 3825 ÷ 3 = 1275, and 1275 ÷ 3 = 425. Next, dividing by 5 gives 425 ÷ 5 = 85 and 85 ÷ 5 = 17. Finally, 17 ÷ 17 = 1.

The number ends in 5, so we divide by 5: 5005 ÷ 5 = 1001. Next, testing subsequent primes: 1001 ÷ 7 = 143, 143 ÷ 11 = 13, and 13 ÷ 13 = 1. Hence, the prime factors are 5 × 7 × 11 × 13.

Testing prime numbers shows that 7429 is not divisible by 2, 3, 5, 7, 11, or 13. Dividing by 17 gives 7429 ÷ 17 = 437. Next, 437 ÷ 19 = 23, and 23 is prime. Thus, 7429 = 17 × 19 × 23.

EXERCISE 1.1 1. Express each number as a product of its prime factors: (i) $140$ (ii) $156$ (iii) $3825$ (iv) $5005$ (v) $7429$ 2. Find the LCM and HCF of the following pairs of integers and verify that $\text{LCM} \times \text{HCF} = \text{product of the two numbers}$ . (i) $26$ and $91$ (ii) $510$ and $92$ (iii) $336$ and $54$ 3. Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) $12, 15$ and $21$ (ii) $17, 23$ and $29$ (iii) $8, 9$ and $25$ 4. Given that $\text{HCF}(306, 657) = 9$ , find $\text{LCM}(306, 657)$ . 5. Check whether $6^n$ can end with the digit $0$ for any natural number $n$ . 6. Explain why $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers.

Answer: (i) $140 = 2^2 \times 5 \times 7$ (ii) $156 = 2^2 \times 3 \times 13$ (iii) $3825 = 3^2 \times 5^2 \times 17$ (iv) $5005 = 5 \times 7 \times 11 \times 13$ (v) $7429 = 17 \times 19 \times 23$

Step-by-step solution

Step 1: Prime factorisation of 140

We divide 140 successively by the smallest possible prime numbers: $140 \div 2 = 70$ , $70 \div 2 = 35$ , $35 \div 5 = 7$ , and $7 \div 7 = 1$ . Thus, the prime factorization of 140 is $2^2 \times 5 \times 7$ .

Step 2: Prime factorisation of 156

Dividing 156 by prime numbers gives $156 \div 2 = 78$ , $78 \div 2 = 39$ , $39 \div 3 = 13$ , and $13 \div 13 = 1$ . Therefore, 156 can be expressed as $2^2 \times 3 \times 13$ .

Step 3: Prime factorisation of 3825

Since the sum of digits is $3 + 8 + 2 + 5 = 18$ , it is divisible by 3: $3825 \div 3 = 1275$ , and $1275 \div 3 = 425$ . Next, dividing by 5 gives $425 \div 5 = 85$ and $85 \div 5 = 17$ . Finally, $17 \div 17 = 1$ .

Step 4: Prime factorisation of 5005

The number ends in 5, so we divide by 5: $5005 \div 5 = 1001$ . Next, testing subsequent primes: $1001 \div 7 = 143$ , $143 \div 11 = 13$ , and $13 \div 13 = 1$ . Hence, the prime factors are $5 \times 7 \times 11 \times 13$ .

Step 5: Prime factorisation of 7429

Testing prime numbers shows that 7429 is not divisible by 2, 3, 5, 7, 11, or 13. Dividing by 17 gives $7429 \div 17 = 437$ . Next, $437 \div 19 = 23$ , and 23 is prime. Thus, $7429 = 17 \times 19 \times 23$ .

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