##### Question 1:

Find the one’s digit of the cube of each of the following numbers.

(i) 3331 (ii) 8888 (iii) 149

(iv) 1005 (v) 1024 (vi) 77

(vii) 5022 (viii) 53

##### Answer:

As we know that, the cube of a number ending in 0, 1, 4, 5, 6 and 9 ends in 0, 1, 4, 5, 6 and 9 respectively. However, the cube of a number ending in 2 ends in 8 and vice-versa. Similarly, the cube of a number ending in 3 or 7 ends in 7 or 3 respectively. Thus, by looking at the unit digit of a given number, we can determine the unit’s digit of its cube. Now, unit’s digit of the cube of each of the asked numbers is as follows:

##### Question 2:

Express the following number as the sum of odd numbers using the above pattern.

6^{3}

##### Answer:

6^{3} = 216 = 31 + 33 + 35 + 37 + 39 + 41

##### Question 3:

Express the following number as the sum of odd numbers using the above pattern.

8^{3}

##### Answer:

8^{3} = 512 = 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71

##### Question 4:

Express the following number as the sum of odd numbers using the above pattern.

7^{3}

##### Answer:

7^{3} = 343 = 43 + 45 + 47 + 49 + 51 + 53 + 55

##### Question 5:

Consider the following pattern:

2^{3} - 1^{3} = 1 + 2 x 1 x 3

3^{3} - 2^{3} = 1 + 3 x 2 x 3

4^{3} - 3^{3} = 1 + 4 x 3 x 3

Using the above pattern, find the value of the
following:

7^{3} - 6^{3}

##### Answer:

7^{3} – 6^{3}

According to the given pattern:

7^{3} – 6^{3} = 1 + 7 x 6 x 3

##### Question 6:

Consider the following pattern:
2^{3} – 1^{3} = 1 + 2 x 1 x 3

3^{3} – 2^{3} = 1 + 3 x 2 x 3

4^{3} – 3^{3} = 1 + 4 x 3 x 3

Using the above pattern, find the value of the

following:

12^{3} – 11^{3}

##### Answer:

12^{3} – 11^{3}

According to the given pattern:

12^{3} – 11^{3} = 1 + 12 x 11 x 3

##### Question 7:

Consider the following pattern:
2^{3} – 1^{3} = 1 + 2 x 1 x 3

3^{3} – 2^{3} = 1 + 3 x 2 x 3

4^{3} – 3^{3} = 1 + 4 x 3 x 3

Using the above pattern, find the value of the
following:

20^{3} – 19^{3}

##### Answer:

20^{3} – 19^{3}

According to the given pattern:

20^{3} – 19^{3} = 1 + 20 x 19 x 3

##### Question 8:

Consider the following pattern:

2^{3} – 1^{3} = 1 + 2 x 1 x 3

3^{3} – 2^{3} = 1 + 3 x 2 x 3

4^{3} – 3^{3} = 1 + 4 x 3 x 3

Using the above pattern, find the value of the
following:

51^{3} – 50^{3}

##### Answer:

51^{3} – 50^{3}

According to the given pattern:

51^{3} – 50^{3} = 1 + 51 x 50 x 3

##### Question 9:

Check whether 400 is perfect cube or not.

##### Answer:

##### Question 10:

Check whether 3375 is perfect cube or not.

##### Answer:

##### Question 11:

Check whether 8000 is perfect cube or not.

##### Answer:

##### Question 12:

Check whether 15625 is perfect cube or not.

##### Answer:

##### Question 13:

Check whether 9000 is perfect cube or not.

##### Answer:

##### Question 14:

Check whether 6859 is perfect cube or not.

##### Answer:

##### Question 15:

Check whether 2025 is perfect cube or not.

##### Answer:

##### Question 16:

Check whether 10648 is perfect cube or not.

##### Answer:

##### Question 17:

Check whether 2700 is perfect cube or not.

##### Answer:

##### Question 18:

Check whether 16000 is perfect cube or not.

##### Answer:

##### Question 19:

Check whether 64000 is perfect cube or not.

##### Answer:

##### Question 20:

Check whether 900 is perfect cube or not.

##### Answer:

##### Question 21:

Check whether 125000 is perfect cube or not.

##### Answer:

##### Question 22:

Check whether 36000 is perfect cube or not.

##### Answer:

##### Question 23:

Check whether 21600 is perfect cube or not.

##### Answer:

##### Question 24:

Check whether 10000 is perfect cube or not.

##### Answer:

##### Question 25:

Check whether 27000000 is perfect cube or not.

##### Answer:

##### Question 26:

Check whether 1000 is perfect cube or not.

##### Answer:

##### Question 27:

Check whether 216 is perfect cube or not.

##### Answer:

##### Question 28:

Check whether 128 is perfect cube or not.

##### Answer:

##### Question 29:

Check whether 1000 is perfect cube or not.

##### Answer:

##### Question 30:

Check whether 100 is perfect cube or not.

##### Answer:

##### Question 31:

Check whether 46656 is perfect cube or not.

##### Answer:

##### Question 32:

Find the smallest number by which 243 must be multiplied to obtain a perfect cube.

##### Answer:

243 = 3 × 3 × 3 × 3 × 3

We note that, prime factor 3 appears in a group of 3. Further, we are left with two more factors 3.

So, if we multiply 3 × 3 by 3 i.e., then in the product 3 will appear in one more group of three and the product will be a perfect cube.
i.e.,

243 × 3 = 3 × 3 × 3 × 3 × 3 × 3 = 729

which is a perfect cube.

Hence, the smallest whole number by which
243 should be multiplied to make a perfect
cube is 3.

##### Question 33:

Find the smallest number by which 256 must be multiplied to obtain a perfect cube.

##### Answer:

256

256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

We note that, prime factor appears in groups of
3. Further, we are left with two more factors 2.

To make it a perfect cube, we need one more 2.

In that case:

256 × 2 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512

which is a perfect cube.

Hence, the smallest whole number by which
256 should be multiplied to make it a perfect
cube is 2.

##### Question 34:

Find the smallest number by which 72 must be multiplied to obtain a perfect cube.

##### Answer:

72

72 = 2 × 2 × 2 × 3 × 3

We note that, prime factor 2 appears in
groups of 3. Further, we are left with two
more factors 3.

To make it a perfect cube, we need one more 3.

In that case:

72 × 3 = 2 × 2 × 2 × 3 × 3 × 3 = 216

which is a perfect cube.

Hence, the smallest whole number by which
72 should be multiplied to make it a perfect
cube is 3.

##### Question 35:

Find the smallest number by which 675 must be multiplied to obtain a perfect cube.

##### Answer:

675

$\therefore $ 675 = 3 × 3 × 3 × 5 × 5

We note that, in prime factorisation of 675,
prime factor 3 appears in a group of 3. Further,
we are left with two more factors 5.

To make it a perfect cube, we need one
more 5.

In that case:

675 × 5 = 3 × 3 × 3 × 5 × 5 × 5 = 3375

which is a perfect cube.

Hence, the smallest whole number by which
675 should be multiplied to make it a perfect
cube is 5.

##### Question 36:

Find the smallest number by which 100 must be multiplied to obtain a perfect cube.

##### Answer:

100

100 = 2 × 2 × 5 × 5

We note that, in the prime factorisation
of 100, the prime factors 2 and 5 are not in
groups of 3.

$\therefore $ We have, one more 2 and one more 5 to
make them in groups of 3.

In that case:

100 × 2 × 5 = 2 × 2 × 2 × 5 × 5 × 5 = 1000

which is a perfect cube.

Hence, the smallest whole number by which
100 should be multiplied to make it a perfect
cube is 10.

##### Question 37:

Find the smallest number by which 81 must be divided to obtain a perfect cube.

##### Answer:

81

$\therefore $ 81 = 3 × 3 × 3 × 3

The prime factor 3 does not appear in a group of three.

$\therefore $ 81 is not a perfect cube.

In the prime factorisation of 81, one factor 3 remains ungrouped.

So, 81 is not a perfect cube.

If we divide 81 by 3, then the prime factorisation
of the quotient becomes 81 ÷ 3 = 3 × 3 × 3 = 27 which is a perfect cube.

Hence, the smallest number by which 81 should be divided to make it a perfect cube is 3.

##### Question 38:

Find the smallest number by which 128 must be divided to obtain a perfect cube.

##### Answer:

128

$\therefore $ 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2

In the prime factorisation of 128, one factor 2 remains ungrouped.

So, 128 is not a perfect cube.

If we divide 128 by 2, then the prime factorisation of the quotient becomes
128 ÷ 2 = 2 × 2 × 2 × 2 × 2 × 2 = 64

which is a perfect cube.

Hence, the smallest number by which 128

should be divided to make it perfect cube is 2.

##### Question 39:

Find the smallest number by which 135 must be divided to obtain a perfect cube.

##### Answer:

135

$\therefore $ 135 = 3 × 3 × 3 × 5

In the prime factorisation of 135, the factor 5 remains ungrouped.

So, 135 is not a perfect cube.

If we divide 135 by 5, then the prime factorisation of the quotient
becomes 135 ÷ 5 = 3 × 3 × 3 = 27

which is a perfect cube.

Hence, the smallest number by which 135 should be divided to make it a perfect cube is 5.

##### Question 40:

Find the smallest number by which 192 must be divided to obtain a perfect cube.192

$\therefore $ 192 = 2 × 2 × 2 × 2 × 2 × 2 × 3

In the prime factorisation of 192, the factor 3 remains ungrouped.

So, 192 is not a perfect cube.

If we divide 192 by 3, then the prime factorisation of the quotient
becomes

192 ÷ 3 = 2 × 2 × 2 × 2× 2× 2 = 64

which is a perfect cube.

Hence, the smallest number by which 192

should be divided to make it a perfect cube is 3.

##### Answer:

192

$\therefore $ 192 = 2 × 2 × 2 × 2 × 2 × 2 × 3

In the prime factorisation of 192, the factor 3 remains ungrouped.

So, 192 is not a perfect cube.

If we divide 192 by 3, then the prime factorisation of the quotient
becomes

192 ÷ 3 = 2 × 2 × 2 × 2× 2× 2 = 64

which is a perfect cube.

Hence, the smallest number by which 192

should be divided to make it a perfect cube is 3.

##### Question 41:

Find the smallest number by which 704 must be divided to obtain a perfect cube.

##### Answer:

704

$\therefore $ 704 = 2 × 2 × 2 × 2 × 2 × 2 × 11

In the prime factorisation of 704, the factor 11 remains ungrouped.

So, 704 is not a perfect cube.

If we divide 704 by 11, then the prime factorisation of the quotient becomes
704 ÷ 11 = 2 × 2 × 2 × 2 × 2 × 2

= 64

which is a perfect cube.

Hence, the smallest number by which 704 should be divided to make it a perfect cube is 11.

##### Question 42:

Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such cuboids will he need to form a cube?

##### Answer:

The sides of cuboid are 5 cm, 2 cm and 5 cm.

Volume of the plastic in = 2 × 5 × 5 cm^{3}

Here, the prime factors of 2 and 5 are not in
groups of three.

$\therefore $ We have to multiply by 2 × 2 × 5 i.e., by 20 to
make it a perfect cube.

##### Question 43:

Find the cube root of 64 by prime factorisation method.

##### Answer:

##### Question 44:

Find the cube root of 512 by prime factorisation method.

##### Answer:

##### Question 45:

Find the cube root of 10648 by prime factorisation method.

##### Answer:

##### Question 46:

Find the cube root of 27000 by prime factorisation method.

##### Answer:

##### Question 47:

Find the cube root of 15625 by prime factorisation method.

##### Answer:

##### Question 48:

Find the cube root of 13824 by prime factorisation method.

##### Answer:

##### Question 49:

Find the cube root of 110592 by prime factorisation method.

##### Answer:

##### Question 50:

Find the cube root of 46656 by prime factorisation method.

##### Answer:

##### Question 51:

Find the cube root of 175616 by prime factorisation method.

##### Answer:

##### Question 52:

Find the cube root of 91125 by prime factorisation method.

##### Answer:

##### Question 53:

Cube of any odd number is even.

- TRUE
- FALSE

##### Answer:

FALSE

##### Question 54:

A perfect cube does not end with two zeros.

- TRUE
- FALSE

##### Answer:

TRUE

##### Question 55:

If square of a number ends with 5, then its cube ends with 25.

- TRUE
- FALSE

##### Answer:

TRUE

##### Question 56:

There is no perfect cube which ends with 8

- TRUE
- FALSE

##### Answer:

FALSE

##### Question 57:

The cube of two digit number may be a three digit number.

- TRUE
- FALSE

##### Answer:

FALSE

##### Question 58:

The cube of a two digit number may have seven or more digits.

- TRUE
- FALSE

##### Answer:

FALSE

##### Question 59:

The cube of a single digit number may be a single digit number.

- TRUE
- FALSE

##### Answer:

TRUE

##### Question 60:

You are told that 1331 is a perfect cube. Can you guess without factorisation what is its cube root? Similarly, guess the cube roots of 4913, 12167, 32768.