Find the greatest number that divides 45 and 240 completely.

Answer

To find the greatest number that divides both 45 and 240 completely, we will determine the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of the two numbers using prime factorization.

### Step-by-Step Solution:

1. **Find the Prime Factorization of 45:**
- Start by dividing 45 by the smallest prime number, which is 3.
- \( 45 \div 3 = 15 \)
- Now, divide 15 by 3 again.
- \( 15 \div 3 = 5 \)
- Finally, 5 is a prime number.
- Therefore, the prime factorization of 45 is:
\[ 45 = 3^2 \times 5^1 \]

2. **Find the Prime Factorization of 240:**
- Start by dividing 240 by the smallest prime number, which is 2.
- \( 240 \div 2 = 120 \)
- Divide 120 by 2.
- \( 120 \div 2 = 60 \)
- Divide 60 by 2.
- \( 60 \div 2 = 30 \)
- Divide 30 by 2.
- \( 30 \div 2 = 15 \)
- Now, divide 15 by 3.
- \( 15 \div 3 = 5 \)
- Finally, 5 is a prime number.
- Therefore, the prime factorization of 240 is:
\[ 240 = 2^4 \times 3^1 \times 5^1 \]

3. **Identify the Common Prime Factors:**
- From the prime factorizations:
- \( 45 = 3^2 \times 5^1 \)
- \( 240 = 2^4 \times 3^1 \times 5^1 \)
- The common prime factors are 3 and 5.

4. **Determine the Lowest Power of Common Factors:**
- For the factor 3, the lowest power is \( 3^1 \).
- For the factor 5, the lowest power is \( 5^1 \).

5. **Calculate the HCF:**
- Multiply the common factors with their lowest powers:
\[ HCF = 3^1 \times 5^1 = 3 \times 5 = 15 \]

### Final Answer:
The greatest number that divides both 45 and 240 completely is **15**.