*Q1: *

*Mention the numbers that are not perfect cubes. *

*(A) 216*

*(B) 128*

*(C) 1000*

*(D) 100*

*(E) 46656*

**Solution:**

(A) 216

Prime factors of 216: 2x2x2x3x3x3

Here all the factors are in the groups of 3’s

Therefore, 216 is said to be a perfect cube number.

02 | 0216 |

02 | 0108 |

02 | 054 |

03 | 027 |

03 | 09 |

03 | 03 |

01 |

(B) 128

The prime factor of 128 = 2x2x2x2x2x2x2

Here one factor 2 does not appear in groups of 3

Hence, 128 is not a perfect cube.

02 | 0128 |

02 | 064 |

02 | 032 |

02 | 016 |

02 | 08 |

02 | 04 |

02 | 02 |

01 |

(C) 1000

The prime factors of 1000 = 2x2x2x 5x5x5

Here all the factors are in groups of 3

Hence, 1000 is said to be a perfect cube.

02 | 01000 |

02 | 0500 |

02 | 0250 |

05 | 0125 |

05 | 025 |

05 | 05 |

01 |

(D) 100

The prime factors of 100 is 2×2 x 5×5

Here all the factors do not appear in groups of 3.

Hence, 100 is not a perfect cube.

02 | 0100 |

02 | 050 |

05 | 025 |

05 | 05 |

01 |

(E) 46656

The prime factors of 46656 = 2x2x2x2x2x2x 3 x3x3x3x3x 3

Here all the factors are in groups of 3

Hence, 46656 is said to be a perfect cube.

02 | 046656 |

02 | 023328 |

02 | 011664 |

02 | 05832 |

02 | 02916 |

02 | 01458 |

03 | 0729 |

03 | 0243 |

03 | 081 |

03 | 027 |

03 | 09 |

03 | 03 |

01 |

*Q2 : *

*Find the smallest number when multiplied to obtain a perfect cube: *

* (A) 243*

*(B) 256*

*(C) 72 *

*(D) 675*

*(E) 100*

**Solution:**

(A) 243

The prime factors of 243 = 3x3x3x3x 3

Here 3 does not appear in groups of 3

Hence, For 243 to be a perfect cube it should be multiplied by 3.

03 | 0243 |

03 | 081 |

03 | 027 |

03 | 09 |

03 | 03 |

01 |

(B) 256

The prime factors of 256 is 2x2x2x2x2x 2 x2 x 2

Here one factor of 2 is required for it to make groups of 3.

Hence, for 256 to be a perfect cube it should be multiplied by 2.

02 | 0256 |

02 | 0128 |

02 | 064 |

02 | 032 |

02 | 016 |

02 | 08 |

02 | 04 |

02 | 02 |

01 |

(C) 72

The prime factors for 72 = 2 x2x2x 3x 3

Here the factor 3 does not appear in groups of 3

Hence, For 72 to be a perfect cube it should be multiplied by 3.

(D) 675

The prime factors for 675 = 3x3x3x 5×5

Here the factor 5 does not appear in groups of 3

Hence, for 675 to be a perfect cube it should be multiplied by 5.

03 | 0675 |

03 | 0225 |

03 | 075 |

05 | 025 |

05 | 05 |

01 |

(E) 100

The prime factors for 100 = 2x2x5x5

Here both the factors 2 and 5 are not in groups of 3

Hence, for 100 to be a perfect cube it should be multiplied by 2 and 5. ( i.e. 2 x 5 =10 )

02 | 0100 |

02 | 050 |

05 | 025 |

05 | 05 |

01 |

*Q3: *

*Find the smallest number by which when divided obtain a perfect cube. *

*(A) 81 *

*(B) 128*

*(C) 135*

*(D) 192 *

*(E) 704 *

**Solution:**

(A) 81

The prime factors for 81 = 3 x 3 x 3 x 3

Here, there is one factor of 3 which extra from the group of 3

Hence, for 81 to be a perfect cube it should be divided by 3.

03 | 081 |

03 | 027 |

03 | 09 |

03 | 03 |

01 |

(B) 128

The prime factors of 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2

Here there is one factor of 2 which in not in the group of 3

Hence, for 128 to be a perfect cube then it should be divided by 2.

02 | 0128 |

02 | 064 |

02 | 032 |

02 | 016 |

02 | 08 |

02 | 04 |

02 | 02 |

01 |

(C) 135

The prime factors of 135 = 3 x 3 x 3 x 5

Here there is one factor of 5 which is not appearing with its group of 3.

Hence, for 135 to be a perfect cube it should be divided by 5.

03 | 0135 |

03 | 045 |

03 | 15 |

05 | 05 |

01 |

(D)192

The prime factors for 192 = 2 x 2 x 2 x 2 x 2 x 2 x 3

Here there is one factor of 3 which does not appearing with its group of 3.

Hence for 192 to be a perfect cube then it should be divided by 3.

02 | 0192 |

02 | 096 |

02 | 048 |

02 | 024 |

02 | 012 |

02 | 06 |

03 | 03 |

01 |

(E) 704

The prime factor for 704 = 2 x 2 x 2 x 2 x 2 x 2 x 11

Here there is one factor of 11 which is not appearing with its group of 3.

Hence for 704 to be a perfect cube it should be divided by 11.

02 | 0704 |

02 | 0352 |

02 | 0176 |

02 | 088 |

02 | 044 |

02 | 022 |

02 | 011 |

01 |

*Q4:*

*Reuben makes a cuboid of clay of sides 5 cm , 2 cm , 5 cm. If Reuben wants to form a cube how many such cuboids will be needed?*

**Solution:**

The numbers given: 5 x 2 x 5

Since the factors of 2 and 4 are both not in groups of 3.

Then, the number should be multiplied by 2 x 2 x 5 = 20 for it to be made a perfect cube.

Hence Reuben needs 20 cuboids.