Gemini on Question 2, Sep 14

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, so we only need to test 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 the divisibility rules and perform division step by step:

• •2: It does not end in an even number.
• •3: The sum of its digits is 9 + 9 + 7 + 3 = 28, which 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 its 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 leaves a remainder when divided by any of the primes up to its square root, it has no prime factors other than 1 and itself. (In fact, 9973 is famously known as the largest four-digit prime number).

[Yes]