AMC 8 Challenge Lab
Lesson 3: Factors, Multiples, Primes & Divisibility
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
Factors, Multiples, Primes & Divisibility
• 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. What is the greatest common factor of 18 and 32?��A. 2 B. 4 C. 8 D. 1 E. 288
2. What is the remainder when 17×21 is divided by 6?��A. 7 B. 3 C. 8 D. 5 E. 4
Practice 3-4
Choose the best answer. Solutions begin after Question 50.
3. What is 17×8−17?��A. 153 B. 136 C. 124 D. 111 E. 119
4. The arithmetic sequence starts 6, 8, 10, ... . What is the 8th term?��A. 18 B. 22 C. 20 D. 16 E. 25
Practice 5-6
Choose the best answer. Solutions begin after Question 50.
5. There are 3 shirt choices, 5 pants choices and 3 hat choices. How many outfits are possible?��A. 45 B. 42 C. 48 D. 11 E. 15
6. What is the greatest common factor of 48 and 32?��A. 22 B. 16 C. 96 D. 8 E. 32
Practice 7-8
Choose the best answer. Solutions begin after Question 50.
7. What is the remainder when 10×15 is divided by 11?��A. 9 B. 10 C. 4 D. 8 E. 7
8. What is 22×18−22?��A. 396 B. 356 C. 418 D. 379 E. 374
Practice 9-10
Choose the best answer. Solutions begin after Question 50.
9. The arithmetic sequence starts 6, 13, 20, ... . What is the 13th term?��A. 83 B. 90 C. 91 D. 97 E. 95
10. There are 3 shirt choices, 6 pants choices and 2 hat choices. How many outfits are possible?��A. 36 B. 11 C. 38 D. 34 E. 18
Practice 11-12
Choose the best answer. Solutions begin after Question 50.
11. What is the greatest common factor of 36 and 32?��A. 2 B. 10 C. 288 D. 4 E. 8
12. What is the remainder when 15×9 is divided by 9?��A. 9 B. 1 C. 0 D. 6 E. 2
Practice 13-14
Choose the best answer. Solutions begin after Question 50.
13. What is 27×13−27?��A. 311 B. 378 C. 324 D. 329 E. 351
14. The arithmetic sequence starts 6, 12, 18, ... . What is the 10th term?��A. 60 B. 64 C. 65 D. 66 E. 54
Practice 15-16
Choose the best answer. Solutions begin after Question 50.
15. There are 3 shirt choices, 7 pants choices and 4 hat choices. How many outfits are possible?��A. 84 B. 21 C. 88 D. 80 E. 14
16. What is the greatest common factor of 24 and 32?��A. 8 B. 14 C. 4 D. 16 E. 96
Practice 17-18
Choose the best answer. Solutions begin after Question 50.
17. What is the remainder when 8×19 is divided by 7?��A. 6 B. 1 C. 0 D. 5 E. 2
18. What is 14×8−14?��A. 103 B. 112 C. 90 D. 126 E. 98
Practice 19-20
Choose the best answer. Solutions begin after Question 50.
19. The arithmetic sequence starts 6, 11, 16, ... . What is the 15th term?��A. 81 B. 75 C. 71 D. 81 E. 76
20. There are 3 shirt choices, 4 pants choices and 3 hat choices. How many outfits are possible?��A. 33 B. 12 C. 39 D. 36 E. 10
Practice 21-22
Choose the best answer. Solutions begin after Question 50.
21. What is the greatest common factor of 54 and 32?��A. 1 B. 864 C. 8 D. 2 E. 4
22. What is the remainder when 13×13 is divided by 5?��A. 0 B. 9 C. 1 D. 3 E. 4
Practice 23-24
Choose the best answer. Solutions begin after Question 50.
23. What is 19×18−19?��A. 328 B. 323 C. 342 D. 361 E. 305
24. The arithmetic sequence starts 6, 10, 14, ... . What is the 12th term?��A. 50 B. 48 C. 55 D. 46 E. 54
Practice 25-26
Choose the best answer. Solutions begin after Question 50.
25. There are 3 shirt choices, 5 pants choices and 2 hat choices. How many outfits are possible?��A. 28 B. 32 C. 30 D. 10 E. 15
26. What is the greatest common factor of 42 and 32?��A. 8 B. 4 C. 672 D. 2 E. 1
Practice 27-28
Choose the best answer. Solutions begin after Question 50.
27. What is the remainder when 18×23 is divided by 10?��A. 9 B. 6 C. 4 D. 5 E. 8
28. What is 24×13−24?��A. 293 B. 312 C. 288 D. 336 E. 275
Practice 29-30
Choose the best answer. Solutions begin after Question 50.
29. The arithmetic sequence starts 6, 9, 12, ... . What is the 9th term?��A. 35 B. 27 C. 33 D. 30 E. 34
30. There are 3 shirt choices, 6 pants choices and 4 hat choices. How many outfits are possible?��A. 68 B. 72 C. 18 D. 76 E. 13
Practice 31-32
Choose the best answer. Solutions begin after Question 50.
31. What is the greatest common factor of 30 and 32?��A. 2 B. 8 C. 480 D. 1 E. 4
32. What is the remainder when 11×17 is divided by 8?��A. 7 B. 3 C. 5 D. 4 E. 8
Practice 33-34
Choose the best answer. Solutions begin after Question 50.
33. What is 29×8−29?��A. 261 B. 232 C. 208 D. 203 E. 195
34. The arithmetic sequence starts 6, 8, 10, ... . What is the 14th term?��A. 37 B. 32 C. 34 D. 30 E. 28
Practice 35-36
Choose the best answer. Solutions begin after Question 50.
35. There are 3 shirt choices, 7 pants choices and 3 hat choices. How many outfits are possible?��A. 60 B. 63 C. 66 D. 21 E. 13
36. What is the greatest common factor of 18 and 32?��A. 8 B. 1 C. 4 D. 288 E. 2
Practice 37-38
Choose the best answer. Solutions begin after Question 50.
37. What is the remainder when 16×11 is divided by 6?��A. 2 B. 4 C. 7 D. 6 E. 3
38. What is 16×18−16?��A. 277 B. 288 C. 254 D. 272 E. 304
Practice 39-40
Choose the best answer. Solutions begin after Question 50.
39. The arithmetic sequence starts 6, 13, 20, ... . What is the 11th term?��A. 77 B. 81 C. 76 D. 69 E. 83
40. There are 3 shirt choices, 4 pants choices and 2 hat choices. How many outfits are possible?��A. 24 B. 26 C. 9 D. 12 E. 22
Practice 41-42
Choose the best answer. Solutions begin after Question 50.
41. What is the greatest common factor of 48 and 32?��A. 16 B. 32 C. 8 D. 22 E. 96
42. What is the remainder when 9×21 is divided by 11?��A. 7 B. 3 C. 9 D. 2 E. 4
Practice 43-44
Choose the best answer. Solutions begin after Question 50.
43. What is 21×13−21?��A. 239 B. 273 C. 257 D. 252 E. 294
44. The arithmetic sequence starts 6, 12, 18, ... . What is the 8th term?��A. 53 B. 52 C. 54 D. 48 E. 42
Practice 45-46
Choose the best answer. Solutions begin after Question 50.
45. There are 3 shirt choices, 5 pants choices and 4 hat choices. How many outfits are possible?��A. 15 B. 60 C. 12 D. 64 E. 56
46. What is the greatest common factor of 36 and 32?��A. 4 B. 2 C. 10 D. 8 E. 288
Practice 47-48
Choose the best answer. Solutions begin after Question 50.
47. What is the remainder when 14×15 is divided by 9?��A. 5 B. 4 C. 8 D. 3 E. 6
48. What is 26×8−26?��A. 208 B. 182 C. 174 D. 187 E. 234
Practice 49-50
Choose the best answer. Solutions begin after Question 50.
49. The arithmetic sequence starts 6, 11, 16, ... . What is the 13th term?��A. 71 B. 65 C. 61 D. 66 E. 71
50. There are 3 shirt choices, 6 pants choices and 3 hat choices. How many outfits are possible?��A. 12 B. 54 C. 51 D. 57 E. 18
Answer Key
Questions 1-50
1. A�2. B�3. E�4. C�5. A�6. B�7. E�8. E�9. B�10. A
11. D�12. C�13. C�14. A�15. A�16. A�17. D�18. E�19. E�20. D
21. D�22. E�23. B�24. A�25. C�26. D�27. C�28. C�29. D�30. B
31. A�32. B�33. D�34. B�35. B�36. E�37. A�38. D�39. C�40. A
41. A�42. D�43. D�44. D�45. B�46. A�47. D�48. B�49. D�50. B
Solutions 1-2
Check the method, not just the letter.
1. Answer: A. 2��Prime-factor or Euclidean-algorithm work gives gcd(18,32)=2.
2. Answer: B. 3��Reduce modulo 6: 17%6=5, 21%6=3; product remainder = 3.
Solutions 3-4
Check the method, not just the letter.
3. Answer: E. 119��Factor 17: 17(8−1)=17×7=119.
4. Answer: C. 20��a_n=a_1+(n−1)d=6+(8−1)(2)=20.
Solutions 5-6
Check the method, not just the letter.
5. Answer: A. 45��Multiplication principle: 3×5×3=45.
6. Answer: B. 16��Prime-factor or Euclidean-algorithm work gives gcd(48,32)=16.
Solutions 7-8
Check the method, not just the letter.
7. Answer: E. 7��Reduce modulo 11: 10%11=10, 15%11=4; product remainder = 7.
8. Answer: E. 374��Factor 22: 22(18−1)=22×17=374.
Solutions 9-10
Check the method, not just the letter.
9. Answer: B. 90��a_n=a_1+(n−1)d=6+(13−1)(7)=90.
10. Answer: A. 36��Multiplication principle: 3×6×2=36.
Solutions 11-12
Check the method, not just the letter.
11. Answer: D. 4��Prime-factor or Euclidean-algorithm work gives gcd(36,32)=4.
12. Answer: C. 0��Reduce modulo 9: 15%9=6, 9%9=0; product remainder = 0.
Solutions 13-14
Check the method, not just the letter.
13. Answer: C. 324��Factor 27: 27(13−1)=27×12=324.
14. Answer: A. 60��a_n=a_1+(n−1)d=6+(10−1)(6)=60.
Solutions 15-16
Check the method, not just the letter.
15. Answer: A. 84��Multiplication principle: 3×7×4=84.
16. Answer: A. 8��Prime-factor or Euclidean-algorithm work gives gcd(24,32)=8.
Solutions 17-18
Check the method, not just the letter.
17. Answer: D. 5��Reduce modulo 7: 8%7=1, 19%7=5; product remainder = 5.
18. Answer: E. 98��Factor 14: 14(8−1)=14×7=98.
Solutions 19-20
Check the method, not just the letter.
19. Answer: E. 76��a_n=a_1+(n−1)d=6+(15−1)(5)=76.
20. Answer: D. 36��Multiplication principle: 3×4×3=36.
Solutions 21-22
Check the method, not just the letter.
21. Answer: D. 2��Prime-factor or Euclidean-algorithm work gives gcd(54,32)=2.
22. Answer: E. 4��Reduce modulo 5: 13%5=3, 13%5=3; product remainder = 4.
Solutions 23-24
Check the method, not just the letter.
23. Answer: B. 323��Factor 19: 19(18−1)=19×17=323.
24. Answer: A. 50��a_n=a_1+(n−1)d=6+(12−1)(4)=50.
Solutions 25-26
Check the method, not just the letter.
25. Answer: C. 30��Multiplication principle: 3×5×2=30.
26. Answer: D. 2��Prime-factor or Euclidean-algorithm work gives gcd(42,32)=2.
Solutions 27-28
Check the method, not just the letter.
27. Answer: C. 4��Reduce modulo 10: 18%10=8, 23%10=3; product remainder = 4.
28. Answer: C. 288��Factor 24: 24(13−1)=24×12=288.
Solutions 29-30
Check the method, not just the letter.
29. Answer: D. 30��a_n=a_1+(n−1)d=6+(9−1)(3)=30.
30. Answer: B. 72��Multiplication principle: 3×6×4=72.
Solutions 31-32
Check the method, not just the letter.
31. Answer: A. 2��Prime-factor or Euclidean-algorithm work gives gcd(30,32)=2.
32. Answer: B. 3��Reduce modulo 8: 11%8=3, 17%8=1; product remainder = 3.
Solutions 33-34
Check the method, not just the letter.
33. Answer: D. 203��Factor 29: 29(8−1)=29×7=203.
34. Answer: B. 32��a_n=a_1+(n−1)d=6+(14−1)(2)=32.
Solutions 35-36
Check the method, not just the letter.
35. Answer: B. 63��Multiplication principle: 3×7×3=63.
36. Answer: E. 2��Prime-factor or Euclidean-algorithm work gives gcd(18,32)=2.
Solutions 37-38
Check the method, not just the letter.
37. Answer: A. 2��Reduce modulo 6: 16%6=4, 11%6=5; product remainder = 2.
38. Answer: D. 272��Factor 16: 16(18−1)=16×17=272.
Solutions 39-40
Check the method, not just the letter.
39. Answer: C. 76��a_n=a_1+(n−1)d=6+(11−1)(7)=76.
40. Answer: A. 24��Multiplication principle: 3×4×2=24.
Solutions 41-42
Check the method, not just the letter.
41. Answer: A. 16��Prime-factor or Euclidean-algorithm work gives gcd(48,32)=16.
42. Answer: D. 2��Reduce modulo 11: 9%11=9, 21%11=10; product remainder = 2.
Solutions 43-44
Check the method, not just the letter.
43. Answer: D. 252��Factor 21: 21(13−1)=21×12=252.
44. Answer: D. 48��a_n=a_1+(n−1)d=6+(8−1)(6)=48.
Solutions 45-46
Check the method, not just the letter.
45. Answer: B. 60��Multiplication principle: 3×5×4=60.
46. Answer: A. 4��Prime-factor or Euclidean-algorithm work gives gcd(36,32)=4.
Solutions 47-48
Check the method, not just the letter.
47. Answer: D. 3��Reduce modulo 9: 14%9=5, 15%9=6; product remainder = 3.
48. Answer: B. 182��Factor 26: 26(8−1)=26×7=182.
Solutions 49-50
Check the method, not just the letter.
49. Answer: D. 66��a_n=a_1+(n−1)d=6+(13−1)(5)=66.
50. Answer: B. 54��Multiplication principle: 3×6×3=54.