Gemini on Question 2, Sep 26

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

The square root of 9973 is approximately 99.86 (since 100² = 10,000). 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 check them step by step:

• •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, 9973 is not divisible by 3.
• •5: The number does not end in 0 or 5, so it is not divisible by 5.
• •7: 9973 ÷ 7 = 1424 with a remainder of 5.
• •11: The alternating sum of the digits is 3 - 7 + 9 - 9 = -4. Since -4 is not divisible by 11, neither is 9973.
• •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 number up to its square root, it has no positive divisors other than 1 and itself. (In fact, 9973 is famously known as the largest four-digit prime number).

[Yes]