Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 65 · 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 units digit of 2^31? (a) 6 (b) 2 (c) 8 (d) 4
2. 2.Which of these numbers is divisible by 4? (a) 755 (b) 746 (c) 748 (d) 750
3. 3.Which of these numbers is divisible by 6? (a) 1330 (b) 1324 (c) 1320 (d) 1316
4. 4.What is the remainder when 188 is divided by 3? (a) 0 (b) 2 (c) 1
5. 5.Which of these numbers is divisible by 11? (a) 7939 (b) 7920 (c) 7912 (d) 7923
6. 6.What is the units digit of 4^24? (a) 7 (b) 5 (c) 4 (d) 6
7. 7.Which of these numbers is divisible by 4? (a) 840 (b) 843 (c) 837 (d) 846
8. 8.What is the smallest number you must add to 622 to make it divisible by 12? (a) 10 (b) 1 (c) 3 (d) 2
9. 9.What is the units digit of 3^33? (a) 1 (b) 7 (c) 3 (d) 9
10. 10.What is the units digit of 7^21? (a) 3 (b) 7 (c) 1 (d) 9
11. 11.What is the remainder when 302 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
12. 12.What is the remainder when 247 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
13. 13.Which of these numbers is divisible by 3? (a) 2222 (b) 2215 (c) 2219 (d) 2217
14. 14.What is the remainder when 86 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
15. 15.What is the units digit of 7^28? (a) 3 (b) 1 (c) 9 (d) 7
16. 16.What is the units digit of 4^51? (a) 3 (b) 5 (c) 6 (d) 4
17. 17.What is the remainder when 109 is divided by 9? (a) 1 (b) 8 (c) 0 (d) 2
18. 18.Which of these numbers is divisible by 6? (a) 3023 (b) 3031 (c) 3027 (d) 3024
19. 19.What is the units digit of 4^53? (a) 6 (b) 4 (c) 3 (d) 5
20. 20.What is the units digit of 3^27? (a) 1 (b) 3 (c) 7 (d) 9
21. 21.Which of these numbers is divisible by 6? (a) 4229 (b) 4217 (c) 4219 (d) 4218
22. 22.What is the smallest number you must add to 517 to make it divisible by 8? (a) 3 (b) 2 (c) 5 (d) 4
23. 23.Which of these numbers is divisible by 6? (a) 1439 (b) 1427 (c) 1428 (d) 1432
24. 24.Which of these numbers is divisible by 6? (a) 2532 (b) 2543 (c) 2527 (d) 2534
25. 25.Which of these numbers is divisible by 8? (a) 4833 (b) 4842 (c) 4850 (d) 4840
26. 26.Which of these numbers is divisible by 8? (a) 3367 (b) 3360 (c) 3354 (d) 3373
27. 27.Which of these numbers is divisible by 4? (a) 1493 (b) 1499 (c) 1502 (d) 1496
28. 28.Which of these numbers is divisible by 6? (a) 4380 (b) 4387 (c) 4384 (d) 4379
29. 29.What is the units digit of 3^13? (a) 7 (b) 9 (c) 1 (d) 3
30. 30.Which of these numbers is divisible by 9? (a) 5043 (b) 5040 (c) 5056 (d) 5037
31. 31.Which of these numbers is divisible by 8? (a) 6393 (b) 6388 (c) 6392 (d) 6405
32. 32.What is the smallest number you must add to 624 to make it divisible by 10? (a) 4 (b) 7 (c) 5 (d) 6
33. 33.What is the units digit of 7^39? (a) 9 (b) 3 (c) 7 (d) 1
34. 34.What is the units digit of 3^10? (a) 3 (b) 1 (c) 9 (d) 7
35. 35.Which of these numbers is divisible by 9? (a) 6264 (b) 6263 (c) 6275 (d) 6269
36. 36.What is the remainder when 82 is divided by 3? (a) 0 (b) 2 (c) 1
37. 37.Which of these numbers is divisible by 8? (a) 2690 (b) 2688 (c) 2702 (d) 2686
38. 38.What is the units digit of 2^33? (a) 4 (b) 2 (c) 8 (d) 6
39. 39.Which of these numbers is divisible by 6? (a) 2244 (b) 2255 (c) 2245 (d) 2242
40. 40.Which of these numbers is divisible by 4? (a) 2353 (b) 2348 (c) 2349 (d) 2345

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