Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 874 · 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.What is the remainder when 49 is divided by 3? (a) 1 (b) 2 (c) 0
2. 2.What is the units digit of 2^33? (a) 4 (b) 6 (c) 2 (d) 8
3. 3.What is the smallest number you must add to 482 to make it divisible by 11? (a) 2 (b) 9 (c) 3 (d) 1
4. 4.What is the smallest number you must add to 394 to make it divisible by 8? (a) 6 (b) 5 (c) 7 (d) 2
5. 5.What is the remainder when 144 is divided by 10? (a) 3 (b) 6 (c) 5 (d) 4
6. 6.Which of these numbers is divisible by 3? (a) 2251 (b) 2247 (c) 2245 (d) 2248
7. 7.What is the units digit of 4^10? (a) 5 (b) 7 (c) 6 (d) 4
8. 8.What is the remainder when 369 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
9. 9.What is the remainder when 51 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
10. 10.Which of these numbers is divisible by 11? (a) 8732 (b) 8712 (c) 8709 (d) 8717
11. 11.What is the remainder when 229 is divided by 8? (a) 6 (b) 4 (c) 5 (d) 3
12. 12.Which of these numbers is divisible by 6? (a) 3465 (b) 3456 (c) 3457 (d) 3454
13. 13.Which of these numbers is divisible by 4? (a) 2894 (b) 2896 (c) 2903 (d) 2897
14. 14.What is the units digit of 8^14? (a) 2 (b) 8 (c) 4 (d) 6
15. 15.What is the remainder when 141 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
16. 16.What is the units digit of 9^40? (a) 0 (b) 2 (c) 9 (d) 1
17. 17.What is the units digit of 9^46? (a) 1 (b) 0 (c) 9 (d) 2
18. 18.Which of these numbers is divisible by 3? (a) 821 (b) 814 (c) 816 (d) 817
19. 19.What is the remainder when 229 is divided by 6? (a) 2 (b) 1 (c) 0 (d) 5
20. 20.What is the smallest number you must add to 320 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
21. 21.What is the smallest number you must add to 561 to make it divisible by 9? (a) 6 (b) 7 (c) 5 (d) 3
22. 22.What is the smallest number you must add to 118 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
23. 23.What is the units digit of 9^47? (a) 7 (b) 9 (c) 8 (d) 1
24. 24.What is the smallest number you must add to 317 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 2
25. 25.What is the remainder when 742 is divided by 11? (a) 7 (b) 4 (c) 5 (d) 6
26. 26.What is the remainder when 562 is divided by 8? (a) 6 (b) 2 (c) 1 (d) 3
27. 27.What is the remainder when 101 is divided by 7? (a) 3 (b) 4 (c) 2 (d) 5
28. 28.Which of these numbers is divisible by 11? (a) 2301 (b) 2310 (c) 2317 (d) 2331
29. 29.Which of these numbers is divisible by 3? (a) 551 (b) 553 (c) 557 (d) 552
30. 30.Which of these numbers is divisible by 11? (a) 8156 (b) 8151 (c) 8166 (d) 8148
31. 31.Which of these numbers is divisible by 11? (a) 8194 (b) 8173 (c) 8164 (d) 8183
32. 32.Which of these numbers is divisible by 3? (a) 2533 (b) 2535 (c) 2537 (d) 2540
33. 33.Which of these numbers is divisible by 11? (a) 6759 (b) 6752 (c) 6743 (d) 6741
34. 34.What is the units digit of 2^52? (a) 4 (b) 8 (c) 2 (d) 6
35. 35.What is the units digit of 4^41? (a) 3 (b) 5 (c) 4 (d) 6
36. 36.Which of these numbers is divisible by 11? (a) 2409 (b) 2403 (c) 2422 (d) 2410
37. 37.Which of these numbers is divisible by 8? (a) 852 (b) 856 (c) 860 (d) 867
38. 38.What is the remainder when 236 is divided by 3? (a) 1 (b) 0 (c) 2
39. 39.Which of these numbers is divisible by 4? (a) 1952 (b) 1949 (c) 1957 (d) 1955
40. 40.What is the smallest number you must add to 96 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4

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