1.    (i) 216

Prime factors of 216 = 2 × 2 × 2 × 3 × 3 × 3

Here all factors are in groups of 3’s (in triples)

Therefore, 216 is a perfect cube number.

(ii) 128

Prime factors of 216 = 2 × 2 × 2 × 2 × 2 × 2 × 2

Here one factor 2 does not appear in a 3’s group.

Therefore, 128 is not a perfect cube.

(iii) 1000

Prime factors of 216 = 2 × 2 × 2 × 3 × 3 × 3

Here all factors appear in 3’s group

Therefore, 1000 s a perfect cube.

(iv) 100

Prime factors of 216 = 2 × 2 × 5 × 5

Here all factors do not appear in a 3’s group.

Therefore, 100 is not a perfect cube.

(v) 46656

Prime factors of 216 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

Here all factors appear in 3’s group

Therefore, 46656 s a perfect cube.

2.    (i) 243

Prime factors of 243 = 3 × 3 × 3 × 3 × 3

Here 3 does not appear in 3’s group.

Therefore, 243 must be multiplied by 3 to make it a perfect cube.

(ii) 256

Prime factors of 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Here one factor 2 is required to make a 3’s group.

Therefore, 256 must be multiplied by 2 to make it a perfect cube.

(iii) 72

Prime factors of 72 = 2 × 2 × 2 × 3 × 3

Here 3 does not appear in 3’s group.

Therefore, 72 must be multiplied by 3 to make it a perfect cube.

(iv) 675

Prime factors of 675 = 3 × 3 × 3 × 5 × 5

Here factor 5 does not appear in 3’s group.

Therefore 675 must be multiplied by 3 to make it a perfect cube.

(v) 100

Prime factors of 100 = 2 × 2 × 5 × 5

Here factor 2 and 5 both do not appear in 3’s group.

Therefore 100 must be multiplied by 2 × 5 = 10 to make it a perfect cube.

3.    (i) 81

Prime factors of 81 = 3 × 3 × 3 × 3

Here one factor 3 is not grouped in triplets.

Therefore 81 must be divided by 3 to make it a perfect cube.

(ii) 128

Prime factors of 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2

Here one factor 2 does not appear in a 3’s group.

Therefore, 128 must be divided by 2 to make it a perfect cube.

(iii) 135

Prime factor of 135 = 3 × 3 × 3 × 5

Here one factor 5 does not appear in a triplet.

Therefore, 135 must be divided by 5 to make it a perfect cube.

(iv) 192

Prime factors of 192 = 2 × 2 × 2 × 2 × 2 × 2 × 3

Here one factor 3 does not appear in a triplet.

Therefore, 192 must be divided by 3 to make it a perfect cube.

(v) 704

Prime factors of 704 = 2 × 2 × 2 × 2 × 2 × 2 × 11

Here one factor 11 does not appear in a triplet.

Therefore, 704 must be divided by 11 to make it a perfect cube.

4.    Given numbers = 5 × 2 × 5

Since, Factors of 5 and 2 both are not in group of three.

Therefore, the number must be multiplied by 2 × 2 × 5 = 20 to make it a perfect cube.

Hence he needs 20 cuboids.

5.    (i) 64

(ii) 512

(iii) 10648

(iii) 27000

(v) 15625

(vi) 13824

(vii) 110592

(viii) 175616

(ix) 91125

6.    (i) False

 Since, 13 = 1, 33 = 27, 53 = 125,………….. are all odd.

(ii)  True

 Since, a perfect cube ends with three zeroes.

e.g. 103 = 1000, 203 = 8000, 30= 27000, ………so on

(iii) False

Since, 52 = 25, 53 = 125, 152 = 225, 153 = 3375

(Did not end with 25)

(iv) False

Since 12= 1728                    [Ends with 8]

And 22= 10648                   [Ends with 8]

(v) False

Since 10= 1728                    [Four digit number]

And 11= 10648                    [Four digit number]

(vi) False

Since 993 = 970299                         [Six digit number]

(vii) True

13 = 1                                                    [Six digit number]

23 = 8                                                   [Single digit number]

7.   We know that 103 = 1000 and possible cube of 113 = 1331

Since, cube of unit’s digit 13 = 1

Therefore, cube root of 1331 is 11.

 

4913

We know that 73 = 343.

Next number comes with 7 as unit place 173 = 4913

Hence, cube root of 4913 is 17.

 

12167

We know that 35 = 27

Here in cube, ones digit is 7

Now next number with 3 as ones digit 135 = 2197

And next number with 3 as ones digit 235 = 12167

Hence cube root of 12167 is 23.

 

32768

We know that 23 = 8

Here in cube, ones digit is 8

Now next number with 2 as ones digit 123 = 1728

And next number with 2 as ones digit 223 = 10648

And next number with 2 as ones digit 323 = 32768

Hence cube root of 32768 is 32.