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**Q.1. ****In which of the following expressions, prime factorization has been done?**

**(i) 24 = 2 × 3 × 4**

**(ii) 56 = 1 × 7 × 2 × 2 × 2**

**(iii) 70 = 2 × 5 × 7**

**(iv) 54 = 2 × 3 × 9**

**Ans: **

**(i)** 24 = 2 × 3 × 4 is not a prime factorisation as 4 is not a prime number.

**(ii)** 56 = 1 × 7 × 2 × 2 × 2 is not a prime factorisation as 1 is not a prime number.

**(iii)** 70 = 2 × 5 × 7 is a prime factorisation as 2, 5, and 7 are prime numbers.

**(iv)** 54 = 2 × 3 × 9 is not a prime factorisation as 9 is not a prime number.

**Q.2. ****Determine prime factorization of each of the following numbers:**

**(i) 216**

**(ii) 420**

**(iii) 468**

**(iv) 945**

**(v) 7325**

**(vi) 13915**

**Ans: **

**(i)** 216

We have:

∴ Prime factorisation of 216 = 2 × 2 × 2 × 3 × 3 × 3

**(ii) **420

We have:

∴ Prime factorisation of 420 = 2 × 2 × 3 × 5 × 7

**(iii)** 468

We have:

∴ Prime factorisation of 468 = 2 × 2 × 3 × 3 × 13

**(iv)** 945

We have:

∴ Prime factorisation of 945 = 3 × 3 × 3 × 5 × 7

**(v) **7325

We have:

∴ Prime factorisation of 7325 = 5 × 5 × 293

**(vi) **13915

We have:

∴ Prime factorisation of 13915 = 5 × 11 × 11 × 23

**Q.3. Write the smallest 4-digit number and express it as a product of primes.**

**Ans: **The smallest 4-digit number is 1000.

1000 = 2 × 500

= 2 × 2 × 250

= 2 × 2 × 2 × 125

= 2 × 2 × 2 × 5 × 25

= 2 × 2 × 2 × 5 × 5 × 5

∴ 1000 = 2 ×2 × 2 × 5 × 5 × 5

**Q.4. Write the largest 4-digit number and give its prime factorization.**

**Ans: **The largest 4-digit number is 9999.

We have:

Hence, the largest 4-digit number 9999 can be expressed in the form of its prime factors as 3 × 3 × 11 × 101.

**Q.5. Find the prime factors of 1729. Arrange the factors in ascending order, and find the relation between two consecutive prime factors.**

**Ans: **The given number is 1729.

We have:

Thus, the number 1729 can be expressed in the form of its prime factors as 7 ×13 ×19.

Relation between its two consecutive prime factors:

The consecutive prime factors of the given number are 7, 13, and 19.

Clearly, 13 − 7 = 6 and 19 − 13 = 6

Here, in two consecutive prime factors, the latter is 6 more than the previous one.

**Q.6. Which factors are not included in the prime factorization of a composite number?**

**Ans: **1 and the number itself are not included in the prime factorisation of a composite number.

Example: 4 is a composite number.

Prime factorisation of 4 = 2 × 2

**Q.7. Here are two different factor trees for 60. Write the missing numbers:**

**(i)**

**(ii)**

**Ans: ****(i)** Since 6 = 2 × 3 and 10 = 5 × 2, we have:

**(ii)** Since 60 = 30 × 2, 30 = 10 × 3 and 10 = 5 × 2, we have:

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