Gemini on Question 2, Sep 22
Is 9973 a prime number? Think step by step and then answer "[Yes]" or "[No]".To determine if 9973 is a prime number, we can check if it is divisible by any prime number up to its square root.
- 1.Find the square root: The square root of 9973 is approximately 99.86 (since $100^2 = 10000$). Therefore, we only need to check for divisibility by prime numbers up to 97.
- 2.List the prime numbers to check: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
- 3.Perform divisibility tests:
- •2: 9973 is an odd number (doesn't end in 0, 2, 4, 6, or 8), so it is not divisible by 2.
- •3: The sum of its digits is $9 + 9 + 7 + 3 = 28$. Since 28 is not divisible by 3, neither is 9973.
- •5: It doesn't end in 0 or 5, so it is not divisible by 5.
- •11: The alternating sum of the digits is $3 - 7 + 9 - 9 = -4$. Since -4 is not divisible by 11, neither is 9973.
- •7, 13, 17, 19... 97: If we divide 9973 by the remaining prime numbers, we will find that none of them divide evenly into 9973. For example:
- •$9973 \div 7 = 1424$ remainder $5$
- •$9973 \div 13 = 767$ remainder $2$
- •$9973 \div 89 = 112$ remainder $5$
- •$9973 \div 97 = 102$ remainder $79$
Since 9973 is not divisible by any prime number less than or equal to its square root, it has no positive divisors other than 1 and itself. (In fact, 9973 is known as the largest four-digit prime number).
[Yes]
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- answered · correct · format ok
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Highlighted: the last [Yes]/[No] bracket, which is the scored answer (Yes).
To determine if 9973 is a prime number, we can check if it is divisible by any prime number up to its square root.
1. **Find the square root:** The square root of 9973 is approximately 99.86 (since $100^2 = 10000$). Therefore, we only need to check for divisibility by prime numbers up to 97.
2. **List the prime numbers to check:** 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
3. **Perform divisibility tests:**
- **2:** 9973 is an odd number (doesn't end in 0, 2, 4, 6, or 8), so it is not divisible by 2.
- **3:** The sum of its digits is $9 + 9 + 7 + 3 = 28$. Since 28 is not divisible by 3, neither is 9973.
- **5:** It doesn't end in 0 or 5, so it is not divisible by 5.
- **11:** The alternating sum of the digits is $3 - 7 + 9 - 9 = -4$. Since -4 is not divisible by 11, neither is 9973.
- **7, 13, 17, 19... 97:** If we divide 9973 by the remaining prime numbers, we will find that none of them divide evenly into 9973. For example:
- $9973 \div 7 = 1424$ remainder $5$
- $9973 \div 13 = 767$ remainder $2$
- $9973 \div 89 = 112$ remainder $5$
- $9973 \div 97 = 102$ remainder $79$
Since 9973 is not divisible by any prime number less than or equal to its square root, it has no positive divisors other than 1 and itself. (In fact, 9973 is known as the largest four-digit prime number).
[Yes]
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