Gemini on Question 2, Sep 25

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

Step 1: Find the square root of 9973.
The square root of 9973 is approximately 99.86. This means we only need to test for divisibility by prime numbers up to 97.
The prime numbers to check are: 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.

Step 2: Test for divisibility.

• •2: 9973 ends in an odd number (3), so it is 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, so it is not divisible by 5.
• •7: 9973 ÷ 7 = 1424 with a remainder of 5.
• •11: 9973 ÷ 11 = 906 with a remainder of 7.
• •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.

Step 3: Conclusion.
Because 9973 cannot be divided evenly by any prime number up to its square root, it has exactly two distinct positive divisors: 1 and itself. (In fact, 9973 is known mathematically as the largest 4-digit prime number).

[Yes]