Which of the following pairs of numbers are co-prime?
215 and 216

Answer

To determine whether the numbers 215 and 216 are co-prime, we need to find their prime factors and check if they have any common factors other than 1.

### Step 1: Prime Factorization of 215
- To find the prime factors of 215, we can start dividing it by the smallest prime numbers.
- 215 is not divisible by 2 (it's odd).
- Next, we check 3: the sum of the digits (2 + 1 + 5 = 8) is not divisible by 3, so 215 is not divisible by 3.
- Now, we check 5: 215 ends with a 5, so it is divisible by 5.
- Dividing 215 by 5 gives us:
\[ 215 \div 5 = 43 \]
- Now, we check if 43 is a prime number. It is not divisible by 2, 3, or 5, confirming that 43 is prime.
- Thus, the prime factorization of 215 is:
\[ 215 = 5 \times 43 \]

### Step 2: Prime Factorization of 216
- Now, we will factor 216.
- 216 is even, so we can divide by 2:
\[ 216 \div 2 = 108 \]
- Continuing to factor 108:
\[ 108 \div 2 = 54 \]
\[ 54 \div 2 = 27 \]
- Now, we factor 27:
\[ 27 \div 3 = 9 \]
\[ 9 \div 3 = 3 \]
\[ 3 \div 3 = 1 \]
- Therefore, the prime factorization of 216 is:
\[ 216 = 2^3 \times 3^3 \]

### Step 3: Comparing the Prime Factors
- The prime factors of 215 are \(5\) and \(43\).
- The prime factors of 216 are \(2^3\) and \(3^3\).
- Now, we check for common factors:
- The factors of 215 (5, 43) do not appear in the factors of 216 (2, 3).

### Conclusion
Since the only common factor between 215 and 216 is 1, we conclude that 215 and 216 are co-prime.

### Final Answer
Yes, 215 and 216 are co-prime numbers.

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