Gemini on Question 2, Sep 19

To determine if 9973 is a prime number, we can check if it is divisible by any prime numbers up to its square root.

• 1.Find the square root: The square root of 9973 is approximately 99.86. Therefore, we only need to check for divisibility by prime numbers up to 97 (which is the largest prime less than 100).
• 2.Test for divisibility by small 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, neither is 9973.
• •5: It 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.
• 3.Test divisibility by remaining primes up to 97:
• •17: 9973 ÷ 17 = 586 remainder 11
• •19: 9973 ÷ 19 = 524 remainder 17
• •23: 9973 ÷ 23 = 433 remainder 14
• •29: 9973 ÷ 29 = 343 remainder 26
• •31: 9973 ÷ 31 = 321 remainder 22
• •37: 9973 ÷ 37 = 269 remainder 20
• •41: 9973 ÷ 41 = 243 remainder 10
• •43: 9973 ÷ 43 = 231 remainder 40
• •47: 9973 ÷ 47 = 212 remainder 9
• •53: 9973 ÷ 53 = 188 remainder 9
• •59: 9973 ÷ 59 = 169 remainder 2
• •61: 9973 ÷ 61 = 163 remainder 30
• •67: 9973 ÷ 67 = 148 remainder 57
• •71: 9973 ÷ 71 = 140 remainder 33
• •73: 9973 ÷ 73 = 136 remainder 45
• •79: 9973 ÷ 79 = 126 remainder 19
• •83: 9973 ÷ 83 = 120 remainder 13
• •89: 9973 ÷ 89 = 112 remainder 5
• •97: 9973 ÷ 97 = 102 remainder 79

Since 9973 cannot be evenly divided by any prime number up to its square root, it only has two distinct positive divisors: 1 and itself. (In fact, 9973 is well known as the largest 4-digit prime number).

[Yes]