Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 330 · 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 3? (a) 1438 (b) 1445 (c) 1440 (d) 1442
2. 2.What is the units digit of 4^20? (a) 6 (b) 5 (c) 7 (d) 4
3. 3.What is the remainder when 130 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
4. 4.Which of these numbers is divisible by 6? (a) 4841 (b) 4844 (c) 4853 (d) 4842
5. 5.What is the units digit of 7^11? (a) 1 (b) 3 (c) 9 (d) 7
6. 6.What is the remainder when 206 is divided by 3? (a) 0 (b) 1 (c) 2
7. 7.What is the smallest number you must add to 202 to make it divisible by 3? (a) 2 (b) 1
8. 8.Which of these numbers is divisible by 3? (a) 1624 (b) 1622 (c) 1619 (d) 1620
9. 9.What is the remainder when 500 is divided by 9? (a) 4 (b) 6 (c) 7 (d) 5
10. 10.Which of these numbers is divisible by 6? (a) 3951 (b) 3954 (c) 3956 (d) 3963
11. 11.What is the smallest number you must add to 293 to make it divisible by 7? (a) 3 (b) 2 (c) 6 (d) 1
12. 12.What is the smallest number you must add to 124 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the units digit of 9^44? (a) 9 (b) 1 (c) 0 (d) 2
14. 14.What is the units digit of 2^38? (a) 6 (b) 8 (c) 2 (d) 4
15. 15.What is the remainder when 468 is divided by 11? (a) 8 (b) 5 (c) 6 (d) 7
16. 16.What is the smallest number you must add to 175 to make it divisible by 11? (a) 2 (b) 3 (c) 1 (d) 10
17. 17.What is the units digit of 4^34? (a) 7 (b) 6 (c) 4 (d) 5
18. 18.What is the remainder when 94 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
19. 19.What is the smallest number you must add to 220 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
20. 20.What is the remainder when 245 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
21. 21.Which of these numbers is divisible by 9? (a) 525 (b) 516 (c) 506 (d) 513
22. 22.What is the remainder when 189 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
23. 23.What is the remainder when 393 is divided by 12? (a) 8 (b) 9 (c) 3 (d) 10
24. 24.What is the smallest number you must add to 149 to make it divisible by 3? (a) 1 (b) 2
25. 25.What is the units digit of 2^47? (a) 4 (b) 2 (c) 6 (d) 8
26. 26.Which of these numbers is divisible by 3? (a) 1990 (b) 1997 (c) 1992 (d) 1994
27. 27.What is the smallest number you must add to 130 to make it divisible by 4? (a) 3 (b) 2 (c) 1
28. 28.Which of these numbers is divisible by 9? (a) 1107 (b) 1113 (c) 1117 (d) 1103
29. 29.What is the remainder when 182 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
30. 30.What is the remainder when 672 is divided by 11? (a) 10 (b) 1 (c) 2 (d) 0
31. 31.Which of these numbers is divisible by 3? (a) 1979 (b) 1976 (c) 1973 (d) 1974
32. 32.What is the units digit of 3^52? (a) 3 (b) 9 (c) 1 (d) 7
33. 33.What is the units digit of 2^32? (a) 4 (b) 2 (c) 6 (d) 8
34. 34.What is the remainder when 462 is divided by 12? (a) 5 (b) 7 (c) 8 (d) 6
35. 35.What is the smallest number you must add to 206 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
36. 36.Which of these numbers is divisible by 6? (a) 579 (b) 569 (c) 574 (d) 570
37. 37.What is the units digit of 2^11? (a) 2 (b) 6 (c) 4 (d) 8
38. 38.What is the units digit of 9^48? (a) 1 (b) 9 (c) 2 (d) 0
39. 39.Which of these numbers is divisible by 6? (a) 2586 (b) 2581 (c) 2591 (d) 2597
40. 40.What is the remainder when 198 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2

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