To determine if 9973 is a prime number, we can check if it has any prime factors less than or equal to its square root.
The square root of 9973 is approximately 99.86. Therefore, we need to check for divisibility by all 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 test a few:
- **2**: It is not even (ends in 3).
- **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 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.
If we continue to divide 9973 by the rest of the prime numbers up to 97 (23, 29, 31... 97), we will find that none of them divide evenly into 9973 (they all leave a remainder). Because it has no positive divisors other than 1 and itself, it is a prime number. (In fact, 9973 is the largest four-digit prime number).
[Yes]