Gemini on Question 2, Sep 21

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

The square root of 9973 is approximately 99.86, so we need to test divisibility for all prime numbers less than 100:
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 an odd number, 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, 9973 is not divisible by 3.
• •5: It does not end in 0 or 5, so it is not divisible by 5.
• •7: 9973 = 7 × 1424 + 5 (remainder is 5)
• •11: 9973 = 11 × 906 + 7 (remainder is 7)
• •13: 9973 = 13 × 767 + 2 (remainder is 2)
• •17: 9973 = 17 × 586 + 11 (remainder is 11)
• •19: 9973 = 19 × 524 + 17 (remainder is 17)
• •23: 9973 = 23 × 433 + 14 (remainder is 14)
• •29: 9973 = 29 × 343 + 26 (remainder is 26)
• •31: 9973 = 31 × 321 + 22 (remainder is 22)
• •37: 9973 = 37 × 269 + 20 (remainder is 20)
• •41: 9973 = 41 × 243 + 10 (remainder is 10)
• •43: 9973 = 43 × 231 + 40 (remainder is 40)
• •47: 9973 = 47 × 212 + 9 (remainder is 9)
• •53: 9973 = 53 × 188 + 9 (remainder is 9)
• •59: 9973 = 59 × 169 + 2 (remainder is 2)
• •61: 9973 = 61 × 163 + 30 (remainder is 30)
• •67: 9973 = 67 × 148 + 57 (remainder is 57)
• •71: 9973 = 71 × 140 + 33 (remainder is 33)
• •73: 9973 = 73 × 136 + 45 (remainder is 45)
• •79: 9973 = 79 × 126 + 19 (remainder is 19)
• •83: 9973 = 83 × 120 + 13 (remainder is 13)
• •89: 9973 = 89 × 112 + 5 (remainder is 5)
• •97: 9973 = 97 × 102 + 79 (remainder is 79)

Because 9973 is not divisible by any prime number up to its square root, it has no divisors other than 1 and itself. (In fact, 9973 is the largest four-digit prime number).

[Yes]