Gemini on Question 2, Sep 11

To determine if 9973 is a prime number, we need to check if it has any positive divisors other than 1 and itself. We can do this by checking for divisibility by all prime numbers up to the square root of 9973.

The square root of 9973 is approximately 99.86 (since 100^2 = 10000). Therefore, we 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 an odd number, so it is not divisible by 2.
• •3: The sum of the digits is 9 + 9 + 7 + 3 = 28. Since 28 is not divisible by 3, neither is 9973.
• •5: 9973 does not end in 0 or 5.
• •7: 9973 ÷ 7 = 1424 with a remainder of 5.
• •11: The alternating sum of the digits is 3 - 7 + 9 - 9 = -4, which is not divisible by 11.
• •13: 9973 ÷ 13 = 767 with a remainder of 2.
• •17: 9973 ÷ 17 = 586 with a remainder of 11.
• •19: 9973 ÷ 19 = 524 with a remainder of 17.
• •23: 9973 ÷ 23 = 433 with a remainder of 14.
• •29: 9973 ÷ 29 = 343 with a remainder of 26.
• •31: 9973 ÷ 31 = 321 with a remainder of 22.
• •37: 9973 ÷ 37 = 269 with a remainder of 20.
• •41: 9973 ÷ 41 = 243 with a remainder of 10.
• •43: 9973 ÷ 43 = 231 with a remainder of 40.
• •47: 9973 ÷ 47 = 212 with a remainder of 9.
• •53: 9973 ÷ 53 = 188 with a remainder of 9.
• •59: 9973 ÷ 59 = 169 with a remainder of 2.
• •61: 9973 ÷ 61 = 163 with a remainder of 30.
• •67: 9973 ÷ 67 = 148 with a remainder of 57.
• •71: 9973 ÷ 71 = 140 with a remainder of 33.
• •73: 9973 ÷ 73 = 136 with a remainder of 45.
• •79: 9973 ÷ 79 = 126 with a remainder of 19.
• •83: 9973 ÷ 83 = 120 with a remainder of 13.
• •89: 9973 ÷ 89 = 112 with a remainder of 5.
• •97: 9973 ÷ 97 = 102 with a remainder of 79.

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

[Yes]