## Checking 9973
**Step 1: Find the limit for testing.**
√9973 ≈ 99.86, so I only need to test prime divisors up to 99.
**Step 2: Test each prime.**
- **2:** 9973 is odd → no
- **3:** digit sum = 9+9+7+3 = 28, not divisible by 3 → no
- **5:** doesn't end in 0 or 5 → no
- **7:** 7 × 1424 = 9968, remainder 5 → no
- **11:** alternating sum = 9−9+7−3 = 4 → no
- **13:** 13 × 767 = 9971, remainder 2 → no
- **17:** 17 × 586 = 9962, remainder 11 → no
- **19:** 19 × 524 = 9956, remainder 17 → no
- **23:** 23 × 433 = 9959, remainder 14 → no
- **29:** 29 × 343 = 9947, remainder 26 → no
- **31:** 31 × 321 = 9951, remainder 22 → no
- **37:** 37 × 269 = 9953, remainder 20 → no
- **41:** 41 × 243 = 9963, remainder 10 → no
- **43:** 43 × 231 = 9933, remainder 40 → no
- **47:** 47 × 212 = 9964, remainder 9 → no
- **53:** 53 × 188 = 9964, remainder 9 → no
- **59:** 59 × 169 = 9971, remainder 2 → no
- **61:** 61 × 163 = 9943, remainder 30 → no
- **67:** 67 × 148 = 9916, remainder 57 → no
- **71:** 71 × 140 = 9940, remainder 33 → no
- **73:** 73 × 136 = 9928, remainder 45 → no
- **79:** 79 × 126 = 9954, remainder 19 → no
- **83:** 83 × 120 = 9960, remainder 13 → no
- **89:** 89 × 112 = 9968, remainder 5 → no
- **97:** 97 × 102 = 9894, remainder 79 → no
**Step 3: Conclusion.**
No prime ≤ 99 divides 9973, so it is prime (in fact, it's the largest four-digit prime).
**[Yes]**