Which of the following pairs of numbers are co-prime numbers for 294 and 256?
The correct answer and explanation is:
Correct Answer: Yes, 294 and 256 are co-prime numbers.
Explanation (300 words):
Co-prime numbers, also known as relatively prime numbers, are two numbers that have no common positive integer factors other than 1. This means that the greatest common divisor (GCD) of the two numbers must be 1.
Let’s analyze the numbers 294 and 256:
Step 1: Find Prime Factors
- 294
- Divide by 2 → 294 ÷ 2 = 147 → 2 is a factor
- 147 ÷ 3 = 49 → 3 is a factor
- 49 = 7 × 7 → 7 is a factor
- Prime factors of 294 = 2, 3, 7
- 256
- 256 is a power of 2: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁸
- Prime factor of 256 = 2
Step 2: Find Common Factors
Now, compare the prime factors:
- 294: 2, 3, 7
- 256: 2
The only common prime factor is 2.
So, are they co-prime?
Since they share the factor 2, they are not co-prime.
🚫 Correction:
Based on the prime factors found, 294 and 256 are NOT co-prime because they both share the prime factor 2.
✅ Final Answer: 294 and 256 are NOT co-prime numbers.
To summarize:
- Co-prime = GCD is 1
- GCD of 294 and 256 = 2
- Hence, they are not co-prime
If you want to find a co-prime pair, look for two numbers with no common factors at all except 1, such as 8 and 15, or 9 and 16.