Gemini on Question 2, Sep 23

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.

Step 1: Find the square root of 9973.
The square root of 9973 is approximately 99.86. Therefore, we only need to check for prime factors up to 97.

Step 2: Check divisibility rules for smaller primes.

• •2: 9973 ends in an odd number (3), so it is not divisible by 2.
• •3: The sum of its digits is 9 + 9 + 7 + 3 = 28. Since 28 is not divisible by 3, 9973 is not divisible by 3.
• •5: 9973 does not end in 0 or 5, so it is not divisible by 5.
• •11: The alternating sum of its digits is 9 - 9 + 7 - 3 = 4. Since 4 is not a multiple of 11, it is not divisible by 11.

Step 3: Test division by remaining primes up to 97.
Let's check the rest of the prime numbers:

• •9973 ÷ 7 = 1424 remainder 5
• •9973 ÷ 13 = 767 remainder 2
• •9973 ÷ 17 = 586 remainder 11
• •9973 ÷ 19 = 524 remainder 17
• •9973 ÷ 23 = 433 remainder 14
• •9973 ÷ 29 = 343 remainder 26
• •9973 ÷ 31 = 321 remainder 22
• •9973 ÷ 37 = 269 remainder 20
• •9973 ÷ 41 = 243 remainder 10
• •9973 ÷ 43 = 231 remainder 40
• •9973 ÷ 47 = 212 remainder 9
• •9973 ÷ 53 = 188 remainder 9
• •9973 ÷ 59 = 169 remainder 2
• •9973 ÷ 61 = 163 remainder 30
• •9973 ÷ 67 = 148 remainder 57
• •9973 ÷ 71 = 140 remainder 33
• •9973 ÷ 73 = 136 remainder 45
• •9973 ÷ 79 = 126 remainder 19
• •9973 ÷ 83 = 120 remainder 13
• •9973 ÷ 89 = 112 remainder 5
• •9973 ÷ 97 = 102 remainder 79

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

[Yes]