#### Exercise: 2.4

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

Answer:

(i) 24 = 2 x 3 x 4 is not a prime factorization as 4 is not a prime number.

(ii) 56 = 1 x 7 x 2 x2 x 2 is not a prime factorization as 1 is not a prime number.

(iii) 70 = 2 x 5 x 7 is a prime factorization as 2, 5, and 7 are prime numbers.

(iv) 54 = 2 x 3 x 9 is not a prime factorization as 9 is not a prime number.

2.) Determine prime factorization of each of the following numbers:

Answer:

(i) 216

We have:

2 | 216 |

2 | 108 |

2 | 54 |

3 | 27 |

3 | 9 |

3 | 3 |

1 |

Therefore, Prime factorization of 216 = 2 x 2 x 2 x3 x 3

(ii) 420

We have:

2 | 420 |

2 | 210 |

3 | 105 |

5 | 35 |

7 | 7 |

1 |

Therefore, Prime factorization of 420= 2 x 2 x 3 x 5 x 7

(iii) 468

We have:

2 | 468 |

2 | 234 |

3 | 117 |

3 | 39 |

13 | 13 |

1 |

Therefore, Prime factorization of 468 = 2 x 2 x 3 x 3 x 13

(iv) 945

We have:

3 | 945 |

3 | 315 |

3 | 105 |

5 | 35 |

7 | 7 |

1 |

Therefore, Prime factorization of 945 = 3 x 3 x 3 x 5 x 7

(v) 7325

We have:

5 | 7325 |

5 | 1465 |

293 | 293 |

1 |

Therefore, Prime factorization of 7325= 5 x 5 x 293

(vi) 13915

We have:

5 | 13915 |

11 | 2783 |

11 | 253 |

23 | 23 |

1 |

Therefore, Prime factorization of 13915 = 5 x 11 x 11 x 23

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

Answer:

The smallest 4-digit number is 1000.

1000 = 2 x 500

=2 x2 x250

=2 x2 x2 x 125

=2 x2x2x5x 25

=2 x2x2x5x5x5

Therefore, 1000=2 x2 x2x5x5x5

4.) Write the largest 4-digit number and express it as product of primes.

Answer:

The largest 4-digit number is 9999.

We have:

3 | 9999 |

3 | 3333 |

11 | 1111 |

101 | 101 |

1 |

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

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

Answer:

The given number is 1729.

We have:

7 | 1729 |

13 | 247 |

19 | 19 |

Thus, the number 1729 can be expressed in the form of its prime factors ass 7 x 13 x19.

Relation between its two consecutive prime factors:

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

Clearly, 13 â€“ 7 = 6 and 19 â€“ 13 =6

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

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

Answer:

1 and the number itself are not included in the prime factorization of a composite number.

Example: 4 is a composite number.

Prime factorization of 4 = 2 x 2.

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

Answer:

(i) Since 6 = 2 x 3 and 10= 5 x 2. We have:

(ii) Since 60 = 30 x 2.

30= 10 x 3 and 10 = 5 x 2 we have: