1. Which of the following numbers are not perfect cubes?
(i) 216 (ii) 128 (iii) 1000 (iv) 100 (v) 46656
(i) 216
Step 1: Prime factorization of 216
| Divisor | Number |
|---|---|
| 2 | 216 |
| 2 | 108 |
| 2 | 54 |
| 3 | 27 |
| 3 | 9 |
| 3 | 3 |
| 1 |
Step 2: Prime factorization expression
\[216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^3 \times 3^3\]
Step 3: Analyze the prime factors
We can see that:
∴ 216 is a perfect cube.
(ii) 128
Step 1: Prime factorization of 128
| Divisor | Number |
|---|---|
| 2 | 128 |
| 2 | 64 |
| 2 | 32 |
| 2 | 16 |
| 2 | 8 |
| 2 | 4 |
| 2 | 2 |
| 1 |
Step 2: Prime factorization expression
\[128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^7\]
Step 3: Analyze the prime factors
We can see that:
∴ 128 is not a perfect cube.
(iii) 1000
Step 1: Prime factorization of 1000
| Divisor | Number |
|---|---|
| 2 | 1000 |
| 2 | 500 |
| 2 | 250 |
| 5 | 125 |
| 5 | 25 |
| 5 | 5 |
| 1 |
Step 2: Prime factorization expression
\[1000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5 = 2^3 \times 5^3\]
Step 3: Analyze the prime factors
We can see that:
∴ 1000 is a perfect cube.
(iv) 100
Step 1: Prime factorization of 100
| Divisor | Number |
|---|---|
| 2 | 100 |
| 2 | 50 |
| 5 | 25 |
| 5 | 5 |
| 1 |
Step 2: Prime factorization expression
\[100 = 2 \times 2 \times 5 \times 5 = 2^2 \times 5^2\]
Step 3: Analyze the prime factors
We can see that:
∴ 100 is not a perfect cube.
(v) 46656
Step 1: Prime factorization of 46656
| Divisor | Number |
|---|---|
| 2 | 46656 |
| 2 | 23328 |
| 2 | 11664 |
| 2 | 5832 |
| 2 | 2916 |
| 2 | 1458 |
| 3 | 729 |
| 3 | 243 |
| 3 | 81 |
| 3 | 27 |
| 3 | 9 |
| 3 | 3 |
| 1 |
Step 2: Prime factorization expression
\[46656 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 2^6 \times 3^6\]
Step 3: Analyze the prime factors
We can see that:
∴ 46656 is a perfect cube.
2. Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.
(i) 243 (ii) 256 (iii) 72 (iv) 675 (v) 100
(i) 243
Step 1: Prime factorization of 243
| Divisor | Number |
|---|---|
| 3 | 243 |
| 3 | 81 |
| 3 | 27 |
| 3 | 9 |
| 3 | 3 |
| 1 |
Step 2: Prime factorization expression
\[243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^3 \times 3^2 \]
Step 3:
To make it a perfect cube, We need to multiply it by 3 .
∴ The smallest number is 3.
(ii) 256
Step 1: Prime factorization of 256
| Divisor | Number |
|---|---|
| 2 | 256 |
| 2 | 128 |
| 2 | 64 |
| 2 | 32 |
| 2 | 16 |
| 2 | 8 |
| 2 | 4 |
| 2 | 2 |
| 1 |
Step 2: Prime factorization expression
\[256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^3 \times 2^3 \times 2^2 \]
Step 3: 2 is without triplet.
To make it a perfect cube, We need to multiply it by 2.
∴ The smallest number is 2.
(iii) 72
Step 1: Prime factorization of 72
| Divisor | Number |
|---|---|
| 2 | 72 |
| 2 | 36 |
| 2 | 18 |
| 3 | 9 |
| 3 | 3 |
| 1 |
Step 2: Prime factorization expression
\[72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2\]
Step 3: 3 is without triplet
To make it a perfect cube, We need to multiply it by 3.
∴ The smallest number is 3.
(iv) 675
Step 1: Prime factorization of 675
| Divisor | Number |
|---|---|
| 3 | 675 |
| 3 | 225 |
| 3 | 75 |
| 5 | 25 |
| 5 | 5 |
| 1 |
Step 2: Prime factorization expression
\[675 = 3 \times 3 \times 3 \times 5 \times 5 = 3^3 \times 5^2\]
Step 3:
To make it a perfect cube, We need to multiply it by 5.
∴ The smallest number is 5.
(v) 100
Step 1: Prime factorization of 100
| Divisor | Number |
|---|---|
| 2 | 100 |
| 2 | 50 |
| 5 | 25 |
| 5 | 5 |
| 1 |
Step 2: Prime factorization expression
\[100 = 2 \times 2 \times 5 \times 5 = 2^2 \times 5^2\]
Step 3: 2 and 5 are without triplet
To make it a perfect cube, We need to multiply it by 2 × 5 = 10
∴ The smallest number is 10.
3. Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube.
(i) 81 (ii) 128 (iii) 135 (iv) 192 (v) 704
(i) 81
Step 1: Prime factorization of 81
| Divisor | Number |
|---|---|
| 3 | 81 |
| 3 | 27 |
| 3 | 9 |
| 3 | 3 |
| 1 |
Step 2: Prime factorization expression
\[81 = 3 \times 3 \times 3 \times 3 = 3^3 \times 3 \]
Step 3: Determine what to divide
To make it a perfect cube, We need to divide it by 3.
∴ The smallest number is 3.
(ii) 128
Step 1: Prime factorization of 128
| Divisor | Number |
|---|---|
| 2 | 128 |
| 2 | 64 |
| 2 | 32 |
| 2 | 16 |
| 2 | 8 |
| 2 | 4 |
| 2 | 2 |
| 1 |
Step 2: Prime factorization expression
\[128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^3 \times 2^3 \times 2 \]
Step 3: Determine what to divide
To make it a perfect cube, We need to divide it by 2.
∴ The smallest number is 2.
(iii) 135
Step 1: Prime factorization of 135
| Divisor | Number |
|---|---|
| 3 | 135 |
| 3 | 45 |
| 3 | 15 |
| 5 | 5 |
| 1 |
Step 2: Prime factorization expression
\[135 = 3 \times 3 \times 3 \times 5 = 3^3 \times 5 \]
Step 3: Determine what to divide
To make it a perfect cube, We need to divide it by 5.
∴ The smallest number is 5.
(iv) 192
Step 1: Prime factorization of 192
| Divisor | Number |
|---|---|
| 2 | 192 |
| 2 | 96 |
| 2 | 48 |
| 2 | 24 |
| 2 | 12 |
| 2 | 6 |
| 3 | 3 |
| 1 |
Step 2: Prime factorization expression
\[192 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 2^6 \times 3 \]
Step 3: Determine what to divide
To make it a perfect cube, We need to divide by 3.
∴ The smallest number is 3.
(v) 704
Step 1: Prime factorization of 704
| Divisor | Number |
|---|---|
| 2 | 704 |
| 2 | 352 |
| 2 | 176 |
| 2 | 88 |
| 2 | 44 |
| 2 | 22 |
| 11 | 11 |
| 1 |
Step 2: Prime factorization expression
\[704 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 11 = 2^6 \times 11 \]
Step 3: Determine what to divide
To make it a perfect cube, We need to divide it by 11.
∴ The smallest number is 11.
4. 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?
Step 1: Find the volume of one cuboid
Volume = length × width × height = 5 cm × 2 cm × 5 cm = 50 cm³
Step 2: Find the prime factors of the volume
50 = 2 × 5 × 5 = 2¹ × 5²
Step 3: Determine what's needed to make it a perfect cube
To make a perfect cube, we need the exponents to be multiples of 3.
We need to multiply by 2² × 5¹ = 4 × 5 = 20
So the volume of the cube would be 50 × 20 = 1000 cm³
Step 4: Calculate how many cuboids are needed
Number of cuboids = Volume of cube ÷ Volume of one cuboid
= 1000 cm³ ÷ 50 cm³ = 20
∴ Parikshit will need 20 such cuboids to form a cube.