AMC 8 Challenge Lab
Lesson 12: Sequences, Patterns & Logical Reasoning
Recognize • Model • Solve • Verify
50 original AMC 8-style multiple-choice problems with complete English solutions.
AMC 8 Strategy
Build speed without sacrificing accuracy.
1. Read for the exact target and hidden restrictions.
2. Translate words into a diagram, equation, table, or smaller case.
3. Estimate before calculating; eliminate impossible choices.
4. Look for shortcuts: symmetry, factoring, parity, ratios, or patterns.
5. Verify the result and make sure it answers the question asked.
Lesson Focus
Sequences, Patterns & Logical Reasoning
• Concept recognition
• Efficient contest strategy
• Mental-math checks
• Multiple representations
• Answer-choice verification
Practice 1-2
Choose the best answer. Solutions begin after Question 50.
1. The arithmetic sequence starts 6, 8, 10, ... . What is the 12th term?��A. 33 B. 24 C. 28 D. 26 E. 30
2. What is 25×12−25?��A. 325 B. 300 C. 275 D. 263 E. 280
Practice 3-4
Choose the best answer. Solutions begin after Question 50.
3. What is the remainder when 9×21 is divided by 7?��A. 0 B. 7 C. 5 D. 2 E. 1
4. There are 5 shirt choices, 7 pants choices and 2 hat choices. How many outfits are possible?��A. 14 B. 35 C. 68 D. 70 E. 72
Practice 5-6
Choose the best answer. Solutions begin after Question 50.
5. If 6x+1=67, what is x?��A. 12 B. 11 C. 7 D. 16 E. 10
6. The arithmetic sequence starts 6, 13, 20, ... . What is the 9th term?��A. 63 B. 55 C. 67 D. 62 E. 69
Practice 7-8
Choose the best answer. Solutions begin after Question 50.
7. What is 12×7−12?��A. 84 B. 65 C. 77 D. 72 E. 96
8. What is the remainder when 14×15 is divided by 5?��A. 2 B. 0 C. 1 D. 5 E. 4
Practice 9-10
Choose the best answer. Solutions begin after Question 50.
9. There are 5 shirt choices, 4 pants choices and 4 hat choices. How many outfits are possible?��A. 76 B. 80 C. 13 D. 20 E. 84
10. If 5x+6=36, what is x?��A. 11 B. 7 C. 6 D. 5 E. 11
Practice 11-12
Choose the best answer. Solutions begin after Question 50.
11. The arithmetic sequence starts 6, 12, 18, ... . What is the 14th term?��A. 78 B. 89 C. 84 D. 88 E. 90
12. What is 17×17−17?��A. 255 B. 306 C. 272 D. 277 E. 289
Practice 13-14
Choose the best answer. Solutions begin after Question 50.
13. What is the remainder when 7×9 is divided by 10?��A. 3 B. 4 C. 8 D. 7 E. 5
14. There are 5 shirt choices, 5 pants choices and 3 hat choices. How many outfits are possible?��A. 13 B. 25 C. 78 D. 75 E. 72
Practice 15-16
Choose the best answer. Solutions begin after Question 50.
15. If 4x+3=47, what is x?��A. 10 B. 11 C. 12 D. 7 E. 16
16. The arithmetic sequence starts 6, 11, 16, ... . What is the 11th term?��A. 55 B. 61 C. 51 D. 61 E. 56
Practice 17-18
Choose the best answer. Solutions begin after Question 50.
17. What is 22×12−22?��A. 242 B. 230 C. 264 D. 247 E. 286
18. What is the remainder when 12×19 is divided by 8?��A. 8 B. 5 C. 6 D. 9 E. 4
Practice 19-20
Choose the best answer. Solutions begin after Question 50.
19. There are 5 shirt choices, 6 pants choices and 2 hat choices. How many outfits are possible?��A. 58 B. 13 C. 30 D. 60 E. 62
20. If 3x+8=26, what is x?��A. 6 B. 5 C. 8 D. 11 E. 7
Practice 21-22
Choose the best answer. Solutions begin after Question 50.
21. The arithmetic sequence starts 6, 10, 14, ... . What is the 8th term?��A. 38 B. 30 C. 32 D. 39 E. 34
22. What is 27×7−27?��A. 162 B. 189 C. 167 D. 155 E. 216
Practice 23-24
Choose the best answer. Solutions begin after Question 50.
23. What is the remainder when 17×13 is divided by 6?��A. 10 B. 5 C. 1 D. 0 E. 9
24. There are 5 shirt choices, 7 pants choices and 4 hat choices. How many outfits are possible?��A. 136 B. 35 C. 16 D. 140 E. 144
Practice 25-26
Choose the best answer. Solutions begin after Question 50.
25. If 2x+5=27, what is x?��A. 10 B. 7 C. 13 D. 11 E. 12
26. The arithmetic sequence starts 6, 9, 12, ... . What is the 13th term?��A. 39 B. 42 C. 46 D. 45 E. 47
Practice 27-28
Choose the best answer. Solutions begin after Question 50.
27. What is 14×17−14?��A. 207 B. 224 C. 229 D. 252 E. 238
28. What is the remainder when 10×23 is divided by 11?��A. 0 B. 10 C. 1 D. 14 E. 15
Practice 29-30
Choose the best answer. Solutions begin after Question 50.
29. There are 5 shirt choices, 4 pants choices and 3 hat choices. How many outfits are possible?��A. 20 B. 57 C. 63 D. 12 E. 60
30. If 7x+2=44, what is x?��A. 6 B. 11 C. 5 D. 9 E. 7
Practice 31-32
Choose the best answer. Solutions begin after Question 50.
31. The arithmetic sequence starts 6, 8, 10, ... . What is the 10th term?��A. 29 B. 26 C. 22 D. 24 E. 20
32. What is 19×12−19?��A. 214 B. 228 C. 247 D. 209 E. 197
Practice 33-34
Choose the best answer. Solutions begin after Question 50.
33. What is the remainder when 15×17 is divided by 9?��A. 6 B. 8 C. 4 D. 3 E. 5
34. There are 5 shirt choices, 5 pants choices and 2 hat choices. How many outfits are possible?��A. 50 B. 48 C. 52 D. 25 E. 12
Practice 35-36
Choose the best answer. Solutions begin after Question 50.
35. If 6x+7=73, what is x?��A. 11 B. 10 C. 13 D. 12 E. 16
36. The arithmetic sequence starts 6, 13, 20, ... . What is the 15th term?��A. 97 B. 105 C. 109 D. 104 E. 111
Practice 37-38
Choose the best answer. Solutions begin after Question 50.
37. What is 24×7−24?��A. 168 B. 149 C. 137 D. 144 E. 192
38. What is the remainder when 8×11 is divided by 7?��A. 5 B. 6 C. 1 D. 4 E. 3
Practice 39-40
Choose the best answer. Solutions begin after Question 50.
39. There are 5 shirt choices, 6 pants choices and 4 hat choices. How many outfits are possible?��A. 116 B. 120 C. 124 D. 30 E. 15
40. If 5x+4=34, what is x?��A. 7 B. 6 C. 5 D. 9 E. 11
Practice 41-42
Choose the best answer. Solutions begin after Question 50.
41. The arithmetic sequence starts 6, 12, 18, ... . What is the 12th term?��A. 66 B. 78 C. 77 D. 72 E. 76
42. What is 29×17−29?��A. 522 B. 493 C. 447 D. 469 E. 464
Practice 43-44
Choose the best answer. Solutions begin after Question 50.
43. What is the remainder when 13×21 is divided by 5?��A. 2 B. 3 C. 4 D. 8 E. 0
44. There are 5 shirt choices, 7 pants choices and 3 hat choices. How many outfits are possible?��A. 15 B. 35 C. 105 D. 102 E. 108
Practice 45-46
Choose the best answer. Solutions begin after Question 50.
45. If 4x+1=45, what is x?��A. 16 B. 10 C. 12 D. 11 E. 5
46. The arithmetic sequence starts 6, 11, 16, ... . What is the 9th term?��A. 51 B. 46 C. 45 D. 51 E. 41
Practice 47-48
Choose the best answer. Solutions begin after Question 50.
47. What is 16×12−16?��A. 176 B. 164 C. 181 D. 208 E. 192
48. What is the remainder when 18×15 is divided by 10?��A. 2 B. 10 C. 1 D. 0 E. 8
Practice 49-50
Choose the best answer. Solutions begin after Question 50.
49. There are 5 shirt choices, 4 pants choices and 2 hat choices. How many outfits are possible?��A. 20 B. 38 C. 42 D. 11 E. 40
50. If 3x+6=24, what is x?��A. 5 B. 9 C. 8 D. 6 E. 7
Answer Key
Questions 1-50
1. C�2. C�3. A�4. D�5. B�6. D�7. D�8. B�9. B�10. C
11. C�12. C�13. A�14. D�15. B�16. E�17. A�18. E�19. D�20. A
21. E�22. A�23. B�24. D�25. D�26. B�27. B�28. B�29. E�30. A
31. D�32. D�33. D�34. A�35. A�36. D�37. D�38. D�39. B�40. B
41. D�42. E�43. B�44. C�45. D�46. B�47. A�48. D�49. E�50. D
Solutions 1-2
Check the method, not just the letter.
1. Answer: C. 28��a_n=a_1+(n−1)d=6+(12−1)(2)=28.
2. Answer: C. 275��Factor 25: 25(12−1)=25×11=275.
Solutions 3-4
Check the method, not just the letter.
3. Answer: A. 0��Reduce modulo 7: 9%7=2, 21%7=0; product remainder = 0.
4. Answer: D. 70��Multiplication principle: 5×7×2=70.
Solutions 5-6
Check the method, not just the letter.
5. Answer: B. 11��Subtract 1, then divide by 6: x=(67−1)/6=11.
6. Answer: D. 62��a_n=a_1+(n−1)d=6+(9−1)(7)=62.
Solutions 7-8
Check the method, not just the letter.
7. Answer: D. 72��Factor 12: 12(7−1)=12×6=72.
8. Answer: B. 0��Reduce modulo 5: 14%5=4, 15%5=0; product remainder = 0.
Solutions 9-10
Check the method, not just the letter.
9. Answer: B. 80��Multiplication principle: 5×4×4=80.
10. Answer: C. 6��Subtract 6, then divide by 5: x=(36−6)/5=6.
Solutions 11-12
Check the method, not just the letter.
11. Answer: C. 84��a_n=a_1+(n−1)d=6+(14−1)(6)=84.
12. Answer: C. 272��Factor 17: 17(17−1)=17×16=272.
Solutions 13-14
Check the method, not just the letter.
13. Answer: A. 3��Reduce modulo 10: 7%10=7, 9%10=9; product remainder = 3.
14. Answer: D. 75��Multiplication principle: 5×5×3=75.
Solutions 15-16
Check the method, not just the letter.
15. Answer: B. 11��Subtract 3, then divide by 4: x=(47−3)/4=11.
16. Answer: E. 56��a_n=a_1+(n−1)d=6+(11−1)(5)=56.
Solutions 17-18
Check the method, not just the letter.
17. Answer: A. 242��Factor 22: 22(12−1)=22×11=242.
18. Answer: E. 4��Reduce modulo 8: 12%8=4, 19%8=3; product remainder = 4.
Solutions 19-20
Check the method, not just the letter.
19. Answer: D. 60��Multiplication principle: 5×6×2=60.
20. Answer: A. 6��Subtract 8, then divide by 3: x=(26−8)/3=6.
Solutions 21-22
Check the method, not just the letter.
21. Answer: E. 34��a_n=a_1+(n−1)d=6+(8−1)(4)=34.
22. Answer: A. 162��Factor 27: 27(7−1)=27×6=162.
Solutions 23-24
Check the method, not just the letter.
23. Answer: B. 5��Reduce modulo 6: 17%6=5, 13%6=1; product remainder = 5.
24. Answer: D. 140��Multiplication principle: 5×7×4=140.
Solutions 25-26
Check the method, not just the letter.
25. Answer: D. 11��Subtract 5, then divide by 2: x=(27−5)/2=11.
26. Answer: B. 42��a_n=a_1+(n−1)d=6+(13−1)(3)=42.
Solutions 27-28
Check the method, not just the letter.
27. Answer: B. 224��Factor 14: 14(17−1)=14×16=224.
28. Answer: B. 10��Reduce modulo 11: 10%11=10, 23%11=1; product remainder = 10.
Solutions 29-30
Check the method, not just the letter.
29. Answer: E. 60��Multiplication principle: 5×4×3=60.
30. Answer: A. 6��Subtract 2, then divide by 7: x=(44−2)/7=6.
Solutions 31-32
Check the method, not just the letter.
31. Answer: D. 24��a_n=a_1+(n−1)d=6+(10−1)(2)=24.
32. Answer: D. 209��Factor 19: 19(12−1)=19×11=209.
Solutions 33-34
Check the method, not just the letter.
33. Answer: D. 3��Reduce modulo 9: 15%9=6, 17%9=8; product remainder = 3.
34. Answer: A. 50��Multiplication principle: 5×5×2=50.
Solutions 35-36
Check the method, not just the letter.
35. Answer: A. 11��Subtract 7, then divide by 6: x=(73−7)/6=11.
36. Answer: D. 104��a_n=a_1+(n−1)d=6+(15−1)(7)=104.
Solutions 37-38
Check the method, not just the letter.
37. Answer: D. 144��Factor 24: 24(7−1)=24×6=144.
38. Answer: D. 4��Reduce modulo 7: 8%7=1, 11%7=4; product remainder = 4.
Solutions 39-40
Check the method, not just the letter.
39. Answer: B. 120��Multiplication principle: 5×6×4=120.
40. Answer: B. 6��Subtract 4, then divide by 5: x=(34−4)/5=6.
Solutions 41-42
Check the method, not just the letter.
41. Answer: D. 72��a_n=a_1+(n−1)d=6+(12−1)(6)=72.
42. Answer: E. 464��Factor 29: 29(17−1)=29×16=464.
Solutions 43-44
Check the method, not just the letter.
43. Answer: B. 3��Reduce modulo 5: 13%5=3, 21%5=1; product remainder = 3.
44. Answer: C. 105��Multiplication principle: 5×7×3=105.
Solutions 45-46
Check the method, not just the letter.
45. Answer: D. 11��Subtract 1, then divide by 4: x=(45−1)/4=11.
46. Answer: B. 46��a_n=a_1+(n−1)d=6+(9−1)(5)=46.
Solutions 47-48
Check the method, not just the letter.
47. Answer: A. 176��Factor 16: 16(12−1)=16×11=176.
48. Answer: D. 0��Reduce modulo 10: 18%10=8, 15%10=5; product remainder = 0.
Solutions 49-50
Check the method, not just the letter.
49. Answer: E. 40��Multiplication principle: 5×4×2=40.
50. Answer: D. 6��Subtract 6, then divide by 3: x=(24−6)/3=6.