Gemini on Question 2, Sep 16

To determine if 9973 is a prime number, we need to check if it is divisible by any prime numbers up to its square root.

The square root of 9973 is approximately 99.86, so we only need to test the prime numbers up to 97:
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.

Let's test them:

• •2: 9973 is odd, so it's not divisible by 2.
• •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 + 5 (not divisible)
• •11: The alternating sum of the digits (3 - 7 + 9 - 9 = -4) is not divisible by 11.
• •13: 9973 = 13 × 767 + 2 (not divisible)
• •17: 9973 = 17 × 586 + 11 (not divisible)
• •19: 9973 = 19 × 524 + 17 (not divisible)
• •23: 9973 = 23 × 433 + 14 (not divisible)
• •29: 9973 = 29 × 343 + 26 (not divisible)
• •31: 9973 = 31 × 321 + 22 (not divisible)
• •37: 9973 = 37 × 269 + 20 (not divisible)
• •41: 9973 = 41 × 243 + 10 (not divisible)
• •43: 9973 = 43 × 231 + 40 (not divisible)
• •47: 9973 = 47 × 212 + 9 (not divisible)
• •53: 9973 = 53 × 188 + 9 (not divisible)
• •59: 9973 = 59 × 169 + 2 (not divisible)
• •61: 9973 = 61 × 163 + 30 (not divisible)
• •67: 9973 = 67 × 148 + 57 (not divisible)
• •71: 9973 = 71 × 140 + 33 (not divisible)
• •73: 9973 = 73 × 136 + 45 (not divisible)
• •79: 9973 = 79 × 126 + 19 (not divisible)
• •83: 9973 = 83 × 120 + 13 (not divisible)
• •89: 9973 = 89 × 112 + 5 (not divisible)
• •97: 9973 = 97 × 102 + 79 (not divisible)

Since 9973 cannot be divided evenly by any prime number up to its square root, it is a prime number. (In fact, it is the largest four-digit prime number).

[Yes]