Gemini on Question 2, Sep 18
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 slightly less than 100 (since 100² = 10,000). Therefore, we need to check for divisibility by all prime numbers up to 97.
- 2.Check basic divisibility rules:
- •2: The number 9973 ends in 3 (an odd number), so it is not divisible by 2.
- •3: The sum of the digits is 9 + 9 + 7 + 3 = 28, which is not divisible by 3.
- •5: The number does not end in 0 or 5, so it is not divisible by 5.
- 3.Check other primes up to 97:
- •7: 9973 ÷ 7 = 1424 remainder 5.
- •11: The alternating sum of digits is 9 - 9 + 7 - 3 = 4, which is not divisible by 11.
- •13: 9973 ÷ 13 = 767 remainder 2.
- •17: 9973 ÷ 17 = 586 remainder 11.
- •19: 9973 ÷ 19 = 524 remainder 17.
- •23: 9973 ÷ 23 = 433 remainder 14.
- •29: 9973 ÷ 29 = 343 remainder 26.
- •31: 9973 ÷ 31 = 321 remainder 22.
- •37: 9973 ÷ 37 = 269 remainder 20.
- •41: 9973 ÷ 41 = 243 remainder 10.
- •43: 9973 ÷ 43 = 231 remainder 40.
- •47: 9973 ÷ 47 = 212 remainder 9.
- •53: 9973 ÷ 53 = 188 remainder 9.
- •59: 9973 ÷ 59 = 169 remainder 2.
- •61: 9973 ÷ 61 = 163 remainder 30.
- •67: 9973 ÷ 67 = 148 remainder 57.
- •71: 9973 ÷ 71 = 140 remainder 33.
- •73: 9973 ÷ 73 = 136 remainder 45.
- •79: 9973 ÷ 79 = 126 remainder 19.
- •83: 9973 ÷ 83 = 120 remainder 13.
- •89: 9973 ÷ 89 = 112 remainder 5.
- •97: 9973 ÷ 97 = 102 remainder 79.
Because 9973 is not divisible by any prime number less than or equal to its square root, it only has two distinct divisors: 1 and itself. (In fact, 9973 is the largest four-digit prime number).
[Yes]
Exact recordShowHideThe model id, the run, the verbatim text with line numbers and its full sha256
- Scored
- answered · correct · format ok
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- 2026-09-18 09:01:34 UTC
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- 20260918T090022Z-92b6f7
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- Google AI Studio · HTTP 200 · finish stop
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- 1,808 characters · 3,780 tokens out incl. hidden reasoning · 36.4 s
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 slightly less than 100 (since 100² = 10,000). Therefore, we need to check for divisibility by all prime numbers up to 97.
2. **Check basic divisibility rules:**
- **2:** The number 9973 ends in 3 (an odd number), so it is not divisible by 2.
- **3:** The sum of the digits is 9 + 9 + 7 + 3 = 28, which is not divisible by 3.
- **5:** The number does not end in 0 or 5, so it is not divisible by 5.
3. **Check other primes up to 97:**
- **7:** 9973 ÷ 7 = 1424 remainder 5.
- **11:** The alternating sum of digits is 9 - 9 + 7 - 3 = 4, which is not divisible by 11.
- **13:** 9973 ÷ 13 = 767 remainder 2.
- **17:** 9973 ÷ 17 = 586 remainder 11.
- **19:** 9973 ÷ 19 = 524 remainder 17.
- **23:** 9973 ÷ 23 = 433 remainder 14.
- **29:** 9973 ÷ 29 = 343 remainder 26.
- **31:** 9973 ÷ 31 = 321 remainder 22.
- **37:** 9973 ÷ 37 = 269 remainder 20.
- **41:** 9973 ÷ 41 = 243 remainder 10.
- **43:** 9973 ÷ 43 = 231 remainder 40.
- **47:** 9973 ÷ 47 = 212 remainder 9.
- **53:** 9973 ÷ 53 = 188 remainder 9.
- **59:** 9973 ÷ 59 = 169 remainder 2.
- **61:** 9973 ÷ 61 = 163 remainder 30.
- **67:** 9973 ÷ 67 = 148 remainder 57.
- **71:** 9973 ÷ 71 = 140 remainder 33.
- **73:** 9973 ÷ 73 = 136 remainder 45.
- **79:** 9973 ÷ 79 = 126 remainder 19.
- **83:** 9973 ÷ 83 = 120 remainder 13.
- **89:** 9973 ÷ 89 = 112 remainder 5.
- **97:** 9973 ÷ 97 = 102 remainder 79.
Because 9973 is not divisible by any prime number less than or equal to its square root, it only has two distinct divisors: 1 and itself. (In fact, 9973 is the largest four-digit prime number).
[Yes]
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