Express each of the following numbers as the sum of twin primes :
(i) 36, (ii) 84, (iii) 120, (iv) 144

Answer

To express each of the given numbers as the sum of twin primes, we will follow a systematic approach. Twin primes are pairs of prime numbers that have a difference of 2.

### Step-by-Step Solution:

#### (i) Expressing 36 as the sum of twin primes:
1. **Divide 36 by 2**:
\[ 36 \div 2 = 18 \]
2. **Find twin primes around 18**:
- Subtract 1: \(18 - 1 = 17\) (which is prime)
- Add 1: \(18 + 1 = 19\) (which is also prime)
3. **Sum of twin primes**:
\[ 17 + 19 = 36 \]

#### (ii) Expressing 84 as the sum of twin primes:
1. **Divide 84 by 2**:
\[ 84 \div 2 = 42 \]
2. **Find twin primes around 42**:
- Subtract 1: \(42 - 1 = 41\) (which is prime)
- Add 1: \(42 + 1 = 43\) (which is also prime)
3. **Sum of twin primes**:
\[ 41 + 43 = 84 \]

#### (iii) Expressing 120 as the sum of twin primes:
1. **Divide 120 by 2**:
\[ 120 \div 2 = 60 \]
2. **Find twin primes around 60**:
- Subtract 1: \(60 - 1 = 59\) (which is prime)
- Add 1: \(60 + 1 = 61\) (which is also prime)
3. **Sum of twin primes**:
\[ 59 + 61 = 120 \]

#### (iv) Expressing 144 as the sum of twin primes:
1. **Divide 144 by 2**:
\[ 144 \div 2 = 72 \]
2. **Find twin primes around 72**:
- Subtract 1: \(72 - 1 = 71\) (which is prime)
- Add 1: \(72 + 1 = 73\) (which is also prime)
3. **Sum of twin primes**:
\[ 71 + 73 = 144 \]

### Final Answers:
- (i) \(36 = 17 + 19\)
- (ii) \(84 = 41 + 43\)
- (iii) \(120 = 59 + 61\)
- (iv) \(144 = 71 + 73\)