Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 645 · 40 questions

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

1. 1.Which of these numbers is divisible by 4? (a) 3581 (b) 3577 (c) 3580 (d) 3585
2. 2.What is the remainder when 311 is divided by 8? (a) 1 (b) 7 (c) 5 (d) 6
3. 3.Which of these numbers is divisible by 11? (a) 4591 (b) 4580 (c) 4576 (d) 4568
4. 4.What is the units digit of 4^19? (a) 4 (b) 6 (c) 3 (d) 5
5. 5.Which of these numbers is divisible by 3? (a) 131 (b) 134 (c) 137 (d) 132
6. 6.What is the smallest number you must add to 131 to make it divisible by 9? (a) 5 (b) 3 (c) 4 (d) 6
7. 7.What is the units digit of 2^20? (a) 4 (b) 8 (c) 2 (d) 6
8. 8.Which of these numbers is divisible by 6? (a) 3017 (b) 3027 (c) 3022 (d) 3018
9. 9.What is the smallest number you must add to 412 to make it divisible by 12? (a) 7 (b) 4 (c) 8 (d) 9
10. 10.What is the units digit of 7^40? (a) 1 (b) 9 (c) 3 (d) 7
11. 11.Which of these numbers is divisible by 8? (a) 842 (b) 859 (c) 849 (d) 848
12. 12.Which of these numbers is divisible by 6? (a) 4156 (b) 4160 (c) 4152 (d) 4149
13. 13.Which of these numbers is divisible by 9? (a) 1418 (b) 1407 (c) 1404 (d) 1396
14. 14.What is the smallest number you must add to 424 to make it divisible by 10? (a) 5 (b) 6 (c) 7 (d) 4
15. 15.What is the smallest number you must add to 349 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 2
16. 16.What is the remainder when 68 is divided by 6? (a) 1 (b) 2 (c) 3 (d) 4
17. 17.What is the units digit of 2^45? (a) 6 (b) 8 (c) 4 (d) 2
18. 18.What is the remainder when 541 is divided by 12? (a) 0 (b) 2 (c) 1 (d) 11
19. 19.What is the smallest number you must add to 257 to make it divisible by 10? (a) 3 (b) 2 (c) 4 (d) 7
20. 20.What is the remainder when 128 is divided by 10? (a) 8 (b) 2 (c) 9 (d) 7
21. 21.What is the remainder when 578 is divided by 12? (a) 2 (b) 10 (c) 3 (d) 1
22. 22.What is the units digit of 4^18? (a) 6 (b) 5 (c) 7 (d) 4
23. 23.Which of these numbers is divisible by 4? (a) 3255 (b) 3251 (c) 3252 (d) 3257
24. 24.What is the remainder when 572 is divided by 9? (a) 5 (b) 4 (c) 7 (d) 6
25. 25.What is the units digit of 7^34? (a) 1 (b) 3 (c) 9 (d) 7
26. 26.What is the units digit of 8^41? (a) 6 (b) 4 (c) 8 (d) 2
27. 27.What is the units digit of 7^24? (a) 7 (b) 9 (c) 3 (d) 1
28. 28.Which of these numbers is divisible by 9? (a) 2680 (b) 2668 (c) 2664 (d) 2656
29. 29.What is the smallest number you must add to 323 to make it divisible by 11? (a) 8 (b) 6 (c) 7 (d) 4
30. 30.Which of these numbers is divisible by 8? (a) 2106 (b) 2104 (c) 2118 (d) 2099
31. 31.What is the units digit of 9^25? (a) 7 (b) 1 (c) 9 (d) 8
32. 32.What is the remainder when 290 is divided by 9? (a) 2 (b) 7 (c) 3 (d) 1
33. 33.What is the smallest number you must add to 67 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
34. 34.What is the smallest number you must add to 840 to make it divisible by 11? (a) 7 (b) 8 (c) 4 (d) 6
35. 35.What is the units digit of 4^45? (a) 4 (b) 5 (c) 3 (d) 6
36. 36.What is the smallest number you must add to 303 to make it divisible by 8? (a) 1 (b) 2 (c) 7 (d) 3
37. 37.What is the units digit of 3^54? (a) 3 (b) 7 (c) 1 (d) 9
38. 38.What is the remainder when 143 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
39. 39.What is the smallest number you must add to 351 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
40. 40.What is the units digit of 3^20? (a) 7 (b) 9 (c) 3 (d) 1

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