Answer :
To determine the greatest possible length for each smaller piece when cutting two pieces of string measuring 240 cm and 168 cm, we need to find the Greatest Common Divisor (GCD) of the two lengths.
Here is a step-by-step method for finding the GCD:
### Step 1: Prime Factorization
First, perform prime factorization on both lengths.
#### Prime Factorization of 240:
1. 240 is even, so divide by 2:
[tex]\( 240 \div 2 = 120 \)[/tex]
2. 120 is even, so divide by 2:
[tex]\( 120 \div 2 = 60 \)[/tex]
3. 60 is even, so divide by 2:
[tex]\( 60 \div 2 = 30 \)[/tex]
4. 30 is even, so divide by 2:
[tex]\( 30 \div 2 = 15 \)[/tex]
5. 15 is divisible by 3 (the next prime number):
[tex]\( 15 \div 3 = 5 \)[/tex]
6. 5 is a prime number.
Thus, the prime factorization of 240 is:
[tex]\( 240 = 2^4 \times 3 \times 5 \)[/tex]
#### Prime Factorization of 168:
1. 168 is even, so divide by 2:
[tex]\( 168 \div 2 = 84 \)[/tex]
2. 84 is even, so divide by 2:
[tex]\( 84 \div 2 = 42 \)[/tex]
3. 42 is even, so divide by 2:
[tex]\( 42 \div 2 = 21 \)[/tex]
4. 21 is divisible by 3 (the next prime number):
[tex]\( 21 \div 3 = 7 \)[/tex]
5. 7 is a prime number.
Thus, the prime factorization of 168 is:
[tex]\( 168 = 2^3 \times 3 \times 7 \)[/tex]
### Step 2: Identify the Common Prime Factors
The prime factorizations of 240 and 168 are:
[tex]\( 240 = 2^4 \times 3 \times 5 \)[/tex]
[tex]\( 168 = 2^3 \times 3 \times 7 \)[/tex]
The common prime factors are 2 and 3. Now, take the smallest exponent for each common prime factor:
- For 2, the smallest exponent is 3 (since [tex]\(2^3\)[/tex] is the smaller power of 2).
- For 3, the smallest exponent is 1.
### Step 3: Calculate the GCD
Multiply the common prime factors using the smallest exponents:
[tex]\[ \text{GCD} = 2^3 \times 3^1 = 8 \times 3 = 24 \][/tex]
Therefore, the greatest possible length for each smaller piece that Rajiv can cut from both string lengths is 24 cm.
Here is a step-by-step method for finding the GCD:
### Step 1: Prime Factorization
First, perform prime factorization on both lengths.
#### Prime Factorization of 240:
1. 240 is even, so divide by 2:
[tex]\( 240 \div 2 = 120 \)[/tex]
2. 120 is even, so divide by 2:
[tex]\( 120 \div 2 = 60 \)[/tex]
3. 60 is even, so divide by 2:
[tex]\( 60 \div 2 = 30 \)[/tex]
4. 30 is even, so divide by 2:
[tex]\( 30 \div 2 = 15 \)[/tex]
5. 15 is divisible by 3 (the next prime number):
[tex]\( 15 \div 3 = 5 \)[/tex]
6. 5 is a prime number.
Thus, the prime factorization of 240 is:
[tex]\( 240 = 2^4 \times 3 \times 5 \)[/tex]
#### Prime Factorization of 168:
1. 168 is even, so divide by 2:
[tex]\( 168 \div 2 = 84 \)[/tex]
2. 84 is even, so divide by 2:
[tex]\( 84 \div 2 = 42 \)[/tex]
3. 42 is even, so divide by 2:
[tex]\( 42 \div 2 = 21 \)[/tex]
4. 21 is divisible by 3 (the next prime number):
[tex]\( 21 \div 3 = 7 \)[/tex]
5. 7 is a prime number.
Thus, the prime factorization of 168 is:
[tex]\( 168 = 2^3 \times 3 \times 7 \)[/tex]
### Step 2: Identify the Common Prime Factors
The prime factorizations of 240 and 168 are:
[tex]\( 240 = 2^4 \times 3 \times 5 \)[/tex]
[tex]\( 168 = 2^3 \times 3 \times 7 \)[/tex]
The common prime factors are 2 and 3. Now, take the smallest exponent for each common prime factor:
- For 2, the smallest exponent is 3 (since [tex]\(2^3\)[/tex] is the smaller power of 2).
- For 3, the smallest exponent is 1.
### Step 3: Calculate the GCD
Multiply the common prime factors using the smallest exponents:
[tex]\[ \text{GCD} = 2^3 \times 3^1 = 8 \times 3 = 24 \][/tex]
Therefore, the greatest possible length for each smaller piece that Rajiv can cut from both string lengths is 24 cm.
