Gemini on Question 2, Oct 2

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 slightly less than 100 (since 100² = 10,000). Therefore, we only need to test for divisibility by 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 do a step-by-step check:

• •2: It does not end in an even digit (ends in 3).
• •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 with a remainder of 5.
• •11: The alternating sum of the digits (3 - 7 + 9 - 9 = -4) 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 evenly divisible by any prime number up to its square root, it cannot be broken down into any smaller whole number factors other than 1 and itself. (In fact, 9973 holds the title for being the largest four-digit prime number).

[Yes]