Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 987 · 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 11? (a) 617 (b) 606 (c) 605 (d) 603
2. 2.Which of these numbers is divisible by 6? (a) 810 (b) 811 (c) 807 (d) 818
3. 3.What is the smallest number you must add to 505 to make it divisible by 11? (a) 3 (b) 1 (c) 2 (d) 10
4. 4.What is the remainder when 563 is divided by 9? (a) 6 (b) 4 (c) 5 (d) 7
5. 5.What is the smallest number you must add to 220 to make it divisible by 8? (a) 6 (b) 5 (c) 3 (d) 4
6. 6.What is the remainder when 682 is divided by 10? (a) 8 (b) 3 (c) 1 (d) 2
7. 7.What is the remainder when 114 is divided by 7? (a) 1 (b) 5 (c) 2 (d) 3
8. 8.What is the remainder when 281 is divided by 12? (a) 5 (b) 6 (c) 7 (d) 4
9. 9.Which of these numbers is divisible by 11? (a) 761 (b) 759 (c) 758 (d) 777
10. 10.What is the units digit of 3^37? (a) 9 (b) 1 (c) 7 (d) 3
11. 11.What is the units digit of 2^33? (a) 6 (b) 8 (c) 2 (d) 4
12. 12.What is the remainder when 92 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
13. 13.What is the remainder when 76 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
14. 14.Which of these numbers is divisible by 9? (a) 929 (b) 938 (c) 936 (d) 953
15. 15.Which of these numbers is divisible by 6? (a) 4359 (b) 4362 (c) 4363 (d) 4370
16. 16.What is the smallest number you must add to 197 to make it divisible by 3? (a) 2 (b) 1
17. 17.What is the remainder when 66 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
18. 18.Which of these numbers is divisible by 3? (a) 127 (b) 130 (c) 126 (d) 125
19. 19.Which of these numbers is divisible by 9? (a) 2124 (b) 2117 (c) 2128 (d) 2137
20. 20.Which of these numbers is divisible by 8? (a) 4883 (b) 4873 (c) 4880 (d) 4895
21. 21.Which of these numbers is divisible by 4? (a) 2700 (b) 2705 (c) 2701 (d) 2697
22. 22.Which of these numbers is divisible by 8? (a) 832 (b) 835 (c) 842 (d) 828
23. 23.What is the units digit of 2^14? (a) 2 (b) 8 (c) 4 (d) 6
24. 24.What is the remainder when 73 is divided by 3? (a) 0 (b) 2 (c) 1
25. 25.Which of these numbers is divisible by 9? (a) 1846 (b) 1854 (c) 1869 (d) 1857
26. 26.What is the smallest number you must add to 588 to make it divisible by 11? (a) 8 (b) 7 (c) 6 (d) 5
27. 27.Which of these numbers is divisible by 9? (a) 710 (b) 725 (c) 711 (d) 719
28. 28.What is the smallest number you must add to 467 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 1
29. 29.What is the units digit of 3^24? (a) 3 (b) 7 (c) 1 (d) 9
30. 30.What is the smallest number you must add to 381 to make it divisible by 11? (a) 5 (b) 7 (c) 4 (d) 3
31. 31.What is the smallest number you must add to 315 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
32. 32.What is the units digit of 8^40? (a) 4 (b) 8 (c) 2 (d) 6
33. 33.What is the smallest number you must add to 133 to make it divisible by 9? (a) 1 (b) 3 (c) 7 (d) 2
34. 34.What is the units digit of 4^16? (a) 7 (b) 6 (c) 4 (d) 5
35. 35.Which of these numbers is divisible by 9? (a) 4770 (b) 4782 (c) 4777 (d) 4769
36. 36.What is the units digit of 3^53? (a) 9 (b) 7 (c) 1 (d) 3
37. 37.Which of these numbers is divisible by 9? (a) 6768 (b) 6783 (c) 6770 (d) 6761
38. 38.Which of these numbers is divisible by 3? (a) 2219 (b) 2215 (c) 2214 (d) 2212
39. 39.What is the smallest number you must add to 65 to make it divisible by 3? (a) 1 (b) 2
40. 40.Which of these numbers is divisible by 3? (a) 2120 (b) 2117 (c) 2114 (d) 2115

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