Gemini on Question 2, Sep 27

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.

• 1.Find the square root of 9973:
  The square root of 9973 is approximately 99.86 (since 100² = 10,000). Therefore, we only need to check prime numbers up to 97.

• 2.Check divisibility by small primes:
• •2: 9973 is odd, 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.

• 3.Check divisibility by larger primes up to 97:
• •7: 9973 ÷ 7 = 1424 with a remainder of 5.
• •11: The alternating sum of digits is 9 - 9 + 7 - 3 = 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.

Since 9973 is not divisible by any prime number less than or equal to its square root, it has no factors other than 1 and itself. (In fact, 9973 is known as the largest four-digit prime number).

[Yes]