AMC 8 Challenge Lab
Lesson 13: Advanced Number Theory & Remainders
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
Advanced Number Theory & Remainders
• 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 remainder when 14×15 is divided by 5?��A. 5 B. 4 C. 1 D. 2 E. 0
2. What is the greatest common factor of 24 and 40?��A. 120 B. 14 C. 16 D. 8 E. 4
Practice 3-4
Choose the best answer. Solutions begin after Question 50.
3. The arithmetic sequence starts 5, 10, 15, ... . What is the 13th term?��A. 65 B. 60 C. 70 D. 69 E. 70
4. What is 16×15−16?��A. 240 B. 229 C. 256 D. 224 E. 209
Practice 5-6
Choose the best answer. Solutions begin after Question 50.
5. If x²−6x+8=0, what is the sum of its roots?��A. 6 B. 11 C. 8 D. 2 E. 4
6. What is the remainder when 7×9 is divided by 10?��A. 3 B. 4 C. 5 D. 7 E. 8
Practice 7-8
Choose the best answer. Solutions begin after Question 50.
7. What is the greatest common factor of 54 and 40?��A. 2 B. 1080 C. 1 D. 8 E. 4
8. The arithmetic sequence starts 5, 9, 13, ... . What is the 10th term?��A. 41 B. 40 C. 46 D. 45 E. 37
Practice 9-10
Choose the best answer. Solutions begin after Question 50.
9. What is 21×10−21?��A. 189 B. 194 C. 179 D. 231 E. 210
10. If x²−6x+8=0, what is the sum of its roots?��A. 2 B. 6 C. 11 D. 4 E. 8
Practice 11-12
Choose the best answer. Solutions begin after Question 50.
11. What is the remainder when 12×19 is divided by 8?��A. 4 B. 9 C. 8 D. 6 E. 5
12. What is the greatest common factor of 42 and 40?��A. 4 B. 840 C. 1 D. 8 E. 2
Practice 13-14
Choose the best answer. Solutions begin after Question 50.
13. The arithmetic sequence starts 5, 8, 11, ... . What is the 15th term?��A. 44 B. 50 C. 52 D. 47 E. 45
14. What is 26×20−26?��A. 520 B. 474 C. 546 D. 494 E. 499
Practice 15-16
Choose the best answer. Solutions begin after Question 50.
15. If x²−6x+8=0, what is the sum of its roots?��A. 6 B. 2 C. 4 D. 8 E. 11
16. What is the remainder when 17×13 is divided by 6?��A. 0 B. 5 C. 10 D. 9 E. 1
Practice 17-18
Choose the best answer. Solutions begin after Question 50.
17. What is the greatest common factor of 30 and 40?��A. 5 B. 20 C. 120 D. 10 E. 16
18. The arithmetic sequence starts 5, 7, 9, ... . What is the 12th term?��A. 29 B. 25 C. 24 D. 32 E. 27
Practice 19-20
Choose the best answer. Solutions begin after Question 50.
19. What is 13×15−13?��A. 182 B. 195 C. 167 D. 208 E. 187
20. If x²−6x+8=0, what is the sum of its roots?��A. 6 B. 2 C. 11 D. 4 E. 8
Practice 21-22
Choose the best answer. Solutions begin after Question 50.
21. What is the remainder when 10×23 is divided by 11?��A. 1 B. 10 C. 0 D. 14 E. 15
22. What is the greatest common factor of 18 and 40?��A. 8 B. 360 C. 1 D. 2 E. 4
Practice 23-24
Choose the best answer. Solutions begin after Question 50.
23. The arithmetic sequence starts 5, 12, 19, ... . What is the 9th term?��A. 66 B. 68 C. 61 D. 54 E. 63
24. What is 18×10−18?��A. 198 B. 152 C. 180 D. 167 E. 162
Practice 25-26
Choose the best answer. Solutions begin after Question 50.
25. If x²−6x+8=0, what is the sum of its roots?��A. 8 B. 2 C. 6 D. 4 E. 11
26. What is the remainder when 15×17 is divided by 9?��A. 8 B. 3 C. 4 D. 5 E. 6
Practice 27-28
Choose the best answer. Solutions begin after Question 50.
27. What is the greatest common factor of 48 and 40?��A. 8 B. 16 C. 240 D. 14 E. 4
28. The arithmetic sequence starts 5, 11, 17, ... . What is the 14th term?��A. 84 B. 83 C. 89 D. 88 E. 77
Practice 29-30
Choose the best answer. Solutions begin after Question 50.
29. What is 23×20−23?��A. 437 B. 417 C. 483 D. 442 E. 460
30. If x²−6x+8=0, what is the sum of its roots?��A. 8 B. 11 C. 6 D. 4 E. 2
Practice 31-32
Choose the best answer. Solutions begin after Question 50.
31. What is the remainder when 8×11 is divided by 7?��A. 4 B. 3 C. 1 D. 6 E. 5
32. What is the greatest common factor of 36 and 40?��A. 360 B. 2 C. 10 D. 8 E. 4
Practice 33-34
Choose the best answer. Solutions begin after Question 50.
33. The arithmetic sequence starts 5, 10, 15, ... . What is the 11th term?��A. 60 B. 50 C. 55 D. 59 E. 60
34. What is 28×15−28?��A. 377 B. 397 C. 392 D. 448 E. 420
Practice 35-36
Choose the best answer. Solutions begin after Question 50.
35. If x²−6x+8=0, what is the sum of its roots?��A. 6 B. 2 C. 4 D. 11 E. 8
36. What is the remainder when 13×21 is divided by 5?��A. 8 B. 3 C. 4 D. 0 E. 2
Practice 37-38
Choose the best answer. Solutions begin after Question 50.
37. What is the greatest common factor of 24 and 40?��A. 120 B. 4 C. 16 D. 8 E. 14
38. The arithmetic sequence starts 5, 9, 13, ... . What is the 8th term?��A. 32 B. 37 C. 38 D. 33 E. 29
Practice 39-40
Choose the best answer. Solutions begin after Question 50.
39. What is 15×10−15?��A. 150 B. 140 C. 125 D. 165 E. 135
40. If x²−6x+8=0, what is the sum of its roots?��A. 6 B. 4 C. 8 D. 11 E. 2
Practice 41-42
Choose the best answer. Solutions begin after Question 50.
41. What is the remainder when 18×15 is divided by 10?��A. 2 B. 8 C. 10 D. 0 E. 1
42. What is the greatest common factor of 54 and 40?��A. 1 B. 2 C. 4 D. 1080 E. 8
Practice 43-44
Choose the best answer. Solutions begin after Question 50.
43. The arithmetic sequence starts 5, 8, 11, ... . What is the 13th term?��A. 39 B. 38 C. 44 D. 46 E. 41
44. What is 20×20−20?��A. 420 B. 380 C. 360 D. 385 E. 400
Practice 45-46
Choose the best answer. Solutions begin after Question 50.
45. If x²−6x+8=0, what is the sum of its roots?��A. 6 B. 11 C. 2 D. 4 E. 8
46. What is the remainder when 11×9 is divided by 8?��A. 7 B. 8 C. 3 D. 5 E. 4
Practice 47-48
Choose the best answer. Solutions begin after Question 50.
47. What is the greatest common factor of 42 and 40?��A. 840 B. 1 C. 4 D. 2 E. 8
48. The arithmetic sequence starts 5, 7, 9, ... . What is the 10th term?��A. 21 B. 28 C. 20 D. 23 E. 25
Practice 49-50
Choose the best answer. Solutions begin after Question 50.
49. What is 25×15−25?��A. 400 B. 355 C. 350 D. 335 E. 375
50. If x²−6x+8=0, what is the sum of its roots?��A. 8 B. 2 C. 11 D. 4 E. 6
Answer Key
Questions 1-50
1. E�2. D�3. A�4. D�5. A�6. A�7. A�8. A�9. A�10. B
11. A�12. E�13. D�14. D�15. A�16. B�17. D�18. E�19. A�20. A
21. B�22. D�23. C�24. E�25. C�26. B�27. A�28. B�29. A�30. C
31. A�32. E�33. C�34. C�35. A�36. B�37. D�38. D�39. E�40. A
41. D�42. B�43. E�44. B�45. A�46. C�47. D�48. D�49. C�50. E
Solutions 1-2
Check the method, not just the letter.
1. Answer: E. 0��Reduce modulo 5: 14%5=4, 15%5=0; product remainder = 0.
2. Answer: D. 8��Prime-factor or Euclidean-algorithm work gives gcd(24,40)=8.
Solutions 3-4
Check the method, not just the letter.
3. Answer: A. 65��a_n=a_1+(n−1)d=5+(13−1)(5)=65.
4. Answer: D. 224��Factor 16: 16(15−1)=16×14=224.
Solutions 5-6
Check the method, not just the letter.
5. Answer: A. 6��For x²−Sx+P=0, the sum of roots is S=6.
6. Answer: A. 3��Reduce modulo 10: 7%10=7, 9%10=9; product remainder = 3.
Solutions 7-8
Check the method, not just the letter.
7. Answer: A. 2��Prime-factor or Euclidean-algorithm work gives gcd(54,40)=2.
8. Answer: A. 41��a_n=a_1+(n−1)d=5+(10−1)(4)=41.
Solutions 9-10
Check the method, not just the letter.
9. Answer: A. 189��Factor 21: 21(10−1)=21×9=189.
10. Answer: B. 6��For x²−Sx+P=0, the sum of roots is S=6.
Solutions 11-12
Check the method, not just the letter.
11. Answer: A. 4��Reduce modulo 8: 12%8=4, 19%8=3; product remainder = 4.
12. Answer: E. 2��Prime-factor or Euclidean-algorithm work gives gcd(42,40)=2.
Solutions 13-14
Check the method, not just the letter.
13. Answer: D. 47��a_n=a_1+(n−1)d=5+(15−1)(3)=47.
14. Answer: D. 494��Factor 26: 26(20−1)=26×19=494.
Solutions 15-16
Check the method, not just the letter.
15. Answer: A. 6��For x²−Sx+P=0, the sum of roots is S=6.
16. Answer: B. 5��Reduce modulo 6: 17%6=5, 13%6=1; product remainder = 5.
Solutions 17-18
Check the method, not just the letter.
17. Answer: D. 10��Prime-factor or Euclidean-algorithm work gives gcd(30,40)=10.
18. Answer: E. 27��a_n=a_1+(n−1)d=5+(12−1)(2)=27.
Solutions 19-20
Check the method, not just the letter.
19. Answer: A. 182��Factor 13: 13(15−1)=13×14=182.
20. Answer: A. 6��For x²−Sx+P=0, the sum of roots is S=6.
Solutions 21-22
Check the method, not just the letter.
21. Answer: B. 10��Reduce modulo 11: 10%11=10, 23%11=1; product remainder = 10.
22. Answer: D. 2��Prime-factor or Euclidean-algorithm work gives gcd(18,40)=2.
Solutions 23-24
Check the method, not just the letter.
23. Answer: C. 61��a_n=a_1+(n−1)d=5+(9−1)(7)=61.
24. Answer: E. 162��Factor 18: 18(10−1)=18×9=162.
Solutions 25-26
Check the method, not just the letter.
25. Answer: C. 6��For x²−Sx+P=0, the sum of roots is S=6.
26. Answer: B. 3��Reduce modulo 9: 15%9=6, 17%9=8; product remainder = 3.
Solutions 27-28
Check the method, not just the letter.
27. Answer: A. 8��Prime-factor or Euclidean-algorithm work gives gcd(48,40)=8.
28. Answer: B. 83��a_n=a_1+(n−1)d=5+(14−1)(6)=83.
Solutions 29-30
Check the method, not just the letter.
29. Answer: A. 437��Factor 23: 23(20−1)=23×19=437.
30. Answer: C. 6��For x²−Sx+P=0, the sum of roots is S=6.
Solutions 31-32
Check the method, not just the letter.
31. Answer: A. 4��Reduce modulo 7: 8%7=1, 11%7=4; product remainder = 4.
32. Answer: E. 4��Prime-factor or Euclidean-algorithm work gives gcd(36,40)=4.
Solutions 33-34
Check the method, not just the letter.
33. Answer: C. 55��a_n=a_1+(n−1)d=5+(11−1)(5)=55.
34. Answer: C. 392��Factor 28: 28(15−1)=28×14=392.
Solutions 35-36
Check the method, not just the letter.
35. Answer: A. 6��For x²−Sx+P=0, the sum of roots is S=6.
36. Answer: B. 3��Reduce modulo 5: 13%5=3, 21%5=1; product remainder = 3.
Solutions 37-38
Check the method, not just the letter.
37. Answer: D. 8��Prime-factor or Euclidean-algorithm work gives gcd(24,40)=8.
38. Answer: D. 33��a_n=a_1+(n−1)d=5+(8−1)(4)=33.
Solutions 39-40
Check the method, not just the letter.
39. Answer: E. 135��Factor 15: 15(10−1)=15×9=135.
40. Answer: A. 6��For x²−Sx+P=0, the sum of roots is S=6.
Solutions 41-42
Check the method, not just the letter.
41. Answer: D. 0��Reduce modulo 10: 18%10=8, 15%10=5; product remainder = 0.
42. Answer: B. 2��Prime-factor or Euclidean-algorithm work gives gcd(54,40)=2.
Solutions 43-44
Check the method, not just the letter.
43. Answer: E. 41��a_n=a_1+(n−1)d=5+(13−1)(3)=41.
44. Answer: B. 380��Factor 20: 20(20−1)=20×19=380.
Solutions 45-46
Check the method, not just the letter.
45. Answer: A. 6��For x²−Sx+P=0, the sum of roots is S=6.
46. Answer: C. 3��Reduce modulo 8: 11%8=3, 9%8=1; product remainder = 3.
Solutions 47-48
Check the method, not just the letter.
47. Answer: D. 2��Prime-factor or Euclidean-algorithm work gives gcd(42,40)=2.
48. Answer: D. 23��a_n=a_1+(n−1)d=5+(10−1)(2)=23.
Solutions 49-50
Check the method, not just the letter.
49. Answer: C. 350��Factor 25: 25(15−1)=25×14=350.
50. Answer: E. 6��For x²−Sx+P=0, the sum of roots is S=6.