Gemini on Question 2, Sep 29
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 numbers up to its square root.
- 1.The square root of 9973 is approximately 99.86. So, we need to test prime numbers up to 99:
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.
- 2.Let's do some quick checks for the smallest primes:
- •2: 9973 is not even.
- •3: The sum of the digits (9 + 9 + 7 + 3 = 28) is not divisible by 3.
- •5: It does not end in 0 or 5.
- •7: 9973 / 7 = 1424 with a remainder of 5.
- •11: The alternating sum of digits (3 - 7 + 9 - 9 = -4) is not a multiple of 11.
- 3.We can continue this division for the rest of the primes up to 97:
- •9973 / 13 = 767 remainder 2
- •9973 / 17 = 586 remainder 11
- •9973 / 19 = 524 remainder 17
- •...
- •9973 / 89 = 112 remainder 5
- •9973 / 97 = 102 remainder 79
Since 9973 cannot be evenly divided by any prime number up to its square root, it has no divisors other than 1 and itself. (In fact, 9973 is well-known in mathematics for being the largest four-digit prime number).
[Yes]
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- answered · correct · format ok
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- 2026-09-29 09:00:56 UTC
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- 20260929T090007Z-334891
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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 numbers up to its square root.
1. The square root of 9973 is approximately 99.86. So, we need to test prime numbers up to 99:
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.
2. Let's do some quick checks for the smallest primes:
- **2**: 9973 is not even.
- **3**: The sum of the digits (9 + 9 + 7 + 3 = 28) is not divisible by 3.
- **5**: It does not end in 0 or 5.
- **7**: 9973 / 7 = 1424 with a remainder of 5.
- **11**: The alternating sum of digits (3 - 7 + 9 - 9 = -4) is not a multiple of 11.
3. We can continue this division for the rest of the primes up to 97:
- 9973 / 13 = 767 remainder 2
- 9973 / 17 = 586 remainder 11
- 9973 / 19 = 524 remainder 17
- ...
- 9973 / 89 = 112 remainder 5
- 9973 / 97 = 102 remainder 79
Since 9973 cannot be evenly divided by any prime number up to its square root, it has no divisors other than 1 and itself. (In fact, 9973 is well-known in mathematics for being the largest four-digit prime number).
[Yes]
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