Using prime factorisation, find which of the following are not perfect cubes.

Options

A. 128
B. 343
C. 729
D. 1331

Answer

To determine which of the given numbers are not perfect cubes using prime factorization, we will follow these steps:

### Step 1: Prime Factorization of 128
1. Start dividing 128 by the smallest prime number, which is 2.
- 128 ÷ 2 = 64
- 64 ÷ 2 = 32
- 32 ÷ 2 = 16
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1

2. Count the number of 2's we have:
- 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2^7

### Step 2: Check if 128 is a Perfect Cube
- A number is a perfect cube if all the prime factors have exponents that are multiples of 3.
- The exponent of 2 in 128 is 7, which is not a multiple of 3.
- Therefore, **128 is not a perfect cube.**

### Step 3: Prime Factorization of 343
1. Start dividing 343 by the smallest prime number, which is 7.
- 343 ÷ 7 = 49
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1

2. Count the number of 7's we have:
- 343 = 7 × 7 × 7 = 7^3

### Step 4: Check if 343 is a Perfect Cube
- The exponent of 7 in 343 is 3, which is a multiple of 3.
- Therefore, **343 is a perfect cube.**

### Step 5: Prime Factorization of 729
1. Start dividing 729 by the smallest prime number, which is 3.
- 729 ÷ 3 = 243
- 243 ÷ 3 = 81
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

2. Count the number of 3's we have:
- 729 = 3 × 3 × 3 × 3 × 3 × 3 = 3^6

### Step 6: Check if 729 is a Perfect Cube
- The exponent of 3 in 729 is 6, which is a multiple of 3.
- Therefore, **729 is a perfect cube.**

### Step 7: Prime Factorization of 1331
1. Start dividing 1331 by the smallest prime number, which is 11.
- 1331 ÷ 11 = 121
- 121 ÷ 11 = 11
- 11 ÷ 11 = 1

2. Count the number of 11's we have:
- 1331 = 11 × 11 × 11 = 11^3

### Step 8: Check if 1331 is a Perfect Cube
- The exponent of 11 in 1331 is 3, which is a multiple of 3.
- Therefore, **1331 is a perfect cube.**

### Conclusion
- The only number that is not a perfect cube is **128**.
- The numbers **343, 729, and 1331** are perfect cubes.