Answer
(a) \(9269\) is composite
(b) \(9103\) is prime
(c) \(8623\) is prime
(d) \(7917\) is composite
Work Step by Step
We will use the “trial‐division” test for primality: If a number \(n\) has a divisor other than 1 and itself, then it must have a prime divisor \(p\) with \(p \le \sqrt{n}\). For each given number, we compute (or estimate) its square root and test divisibility by all primes less than or equal to that bound.
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### (a) \(9269\)
- **Estimate \(\sqrt{9269}\):**
\(96^2 = 9216\) and \(97^2 = 9409\), so \(\sqrt{9269}\) is between 96 and 97.
- **Test for divisibility:**
- **Divisible by 2?** \(9269\) is odd.
- **Divisible by 3?** Sum of digits: \(9+2+6+9=26\); since 26 is not divisible by 3, \(9269\) is not divisible by 3.
- **Divisible by 5?** Last digit is not 0 or 5.
- **Divisible by 7?**
\(7 \times 1324 = 9268\), so \(9269\) gives remainder 1 when divided by 7.
- **Divisible by 11?**
The alternating sum: \((9+6) - (2+9) = 15 - 11 = 4\); not divisible by 11.
- **Divisible by 13?**
Try: \(13 \times 713 = 13 \times 700 + 13 \times 13 = 9100 + 169 = 9269\).
Since \(9269 = 13 \times 713\), it is **composite**.
---
### (b) \(9103\)
- **Estimate \(\sqrt{9103}\):**
\(95^2 = 9025\) and \(96^2 = 9216\), so \(\sqrt{9103}\) is between 95 and 96.
- **Test for divisibility by primes up to 95:**
1. **2:** \(9103\) is odd.
2. **3:** Sum of digits: \(9+1+0+3=13\); not divisible by 3.
3. **5:** Last digit is not 0 or 5.
4. **7:** \(7 \times 1300 = 9100\) with remainder \(9103-9100=3\).
5. **11:** \((9+0) - (1+3)=9-4=5\); not divisible by 11.
6. **13:** \(13 \times 700 = 9100\) with remainder 3.
7. **17:** \(17 \times 535 = 9095\) with remainder 8.
8. **19:** \(19 \times 479 = 9101\) with remainder 2.
9. **23:** \(23 \times 395 = 9085\) (remainder 18) or \(23 \times 396 = 9108\) (overshoots).
10. **29:** \(29 \times 313 = 9067\) (remainder 36) or \(29 \times 314 = 9106\) (overshoots).
11. **31:** \(31 \times 293 = 9083\) (remainder 20).
12. **37:** \(37 \times 246 = 9102\) (remainder 1).
13. **41:** \(41 \times 222 = 9102\) (remainder 1).
14. **43:** \(43 \times 211 = 9073\) (remainder 30).
15. **47:** \(47 \times 193 = 9071\) (remainder 32).
16. **53:** \(53 \times 171 = 9063\) (remainder 40).
17. **59:** \(59 \times 154 = 9086\) (remainder 17).
18. **61:** \(61 \times 149 = 9089\) (remainder 14).
19. **67:** \(67 \times 135 = 9045\) (remainder 58) or \(67 \times 136 = 9112\) (overshoots).
20. **71:** \(71 \times 128 = 9088\) (remainder 15).
21. **73:** \(73 \times 124 = 9052\) (remainder 51) or \(73 \times 125 = 9125\) (overshoots).
22. **79:** \(79 \times 115 = 9085\) (remainder 18).
23. **83:** \(83 \times 109 = 9047\) (remainder 56).
24. **89:** \(89 \times 102 = 9078\) (remainder 25).
None of these primes divide \(9103\) evenly. Thus, \(9103\) is **prime**.
---
### (c) \(8623\)
- **Estimate \(\sqrt{8623}\):**
\(92^2 = 8464\) and \(93^2 = 8649\), so \(\sqrt{8623}\) is between 92 and 93.
- **Test for divisibility by primes up to 92:**
1. **2:** \(8623\) is odd.
2. **3:** Sum of digits: \(8+6+2+3 = 19\); not divisible by 3.
3. **5:** Last digit is not 0 or 5.
4. **7:** \(7 \times 1232 = 8624\); remainder \(8624 - 8623 = 1\).
5. **11:** Using the alternating sum test: \((8+2)-(6+3)=10-9=1\); not divisible by 11.
6. **13:** \(13 \times 663 = 8619\); remainder \(4\).
7. **17:** \(17 \times 507 = 8619\); remainder \(4\).
8. **19:** \(19 \times 453 = 8607\); remainder \(16\).
9. **23:** \(23 \times 375 = 8625\); remainder \(-2\).
10. **29:** \(29 \times 297 = 8613\); remainder \(10\).
11. **31:** \(31 \times 278 = 8618\); remainder \(5\).
12. **37:** \(37 \times 233 = 8601\); remainder \(22\).
13. **41:** \(41 \times 210 = 8610\); remainder \(13\).
14. **43:** \(43 \times 200 = 8600\); remainder \(23\).
15. **47:** \(47 \times 183 = 8601\); remainder \(22\).
16. **53:** \(53 \times 162 = 8586\); remainder \(37\).
17. **59:** \(59 \times 146 = 8614\); remainder \(9\).
18. **61:** \(61 \times 141 = 8601\); remainder \(22\).
19. **67:** \(67 \times 128 = 8576\); remainder \(47\).
20. **71:** \(71 \times 111 = 7881\) is too low; more appropriately, \(71 \times 121 = 8591\); remainder \(32\).
21. **73:** \(73 \times 118 = 8614\); remainder \(9\).
22. **79:** \(79 \times 109 = 8611\); remainder \(12\).
23. **83:** \(83 \times 104 = 8632\); remainder \(-9\).
24. **89:** \(89 \times 96 = 8544\); remainder \(8623-8544=79\).
No prime divisor was found. Hence, \(8623\) is **prime**.
---
### (d) \(7917\)
- **Estimate \(\sqrt{7917}\):**
\(88^2 = 7744\) and \(89^2 = 7921\), so \(\sqrt{7917}\) is a bit less than 89.
- **Test for divisibility:**
1. **2:** \(7917\) is odd.
2. **3:** Sum of digits: \(7+9+1+7 = 24\), which is divisible by 3.
Since \(24\) is divisible by 3, \(7917\) is divisible by 3. In fact,
\[
7917 \div 3 = 2639.
\]
Thus, \(7917\) is **composite**.
