Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 358 · 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 8? (a) 1785 (b) 1784 (c) 1781 (d) 1796
2. 2.What is the smallest number you must add to 244 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
3. 3.What is the remainder when 478 is divided by 9? (a) 8 (b) 2 (c) 1 (d) 0
4. 4.Which of these numbers is divisible by 9? (a) 3303 (b) 3295 (c) 3313 (d) 3307
5. 5.What is the units digit of 7^39? (a) 1 (b) 9 (c) 7 (d) 3
6. 6.What is the smallest number you must add to 91 to make it divisible by 3? (a) 2 (b) 1
7. 7.Which of these numbers is divisible by 11? (a) 4189 (b) 4170 (c) 4168 (d) 4169
8. 8.What is the smallest number you must add to 35 to make it divisible by 3? (a) 1 (b) 2
9. 9.What is the remainder when 539 is divided by 12? (a) 10 (b) 11 (c) 9 (d) 1
10. 10.What is the smallest number you must add to 148 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the units digit of 8^12? (a) 8 (b) 2 (c) 4 (d) 6
12. 12.What is the units digit of 2^21? (a) 6 (b) 8 (c) 4 (d) 2
13. 13.What is the smallest number you must add to 506 to make it divisible by 8? (a) 5 (b) 2 (c) 7 (d) 6
14. 14.What is the remainder when 201 is divided by 8? (a) 7 (b) 0 (c) 1 (d) 2
15. 15.Which of these numbers is divisible by 8? (a) 3500 (b) 3490 (c) 3496 (d) 3505
16. 16.What is the smallest number you must add to 629 to make it divisible by 9? (a) 8 (b) 3 (c) 1 (d) 2
17. 17.What is the smallest number you must add to 284 to make it divisible by 10? (a) 6 (b) 7 (c) 5 (d) 4
18. 18.Which of these numbers is divisible by 9? (a) 2046 (b) 2034 (c) 2029 (d) 2040
19. 19.Which of these numbers is divisible by 11? (a) 4697 (b) 4715 (c) 4688 (d) 4698
20. 20.What is the units digit of 7^52? (a) 3 (b) 1 (c) 9 (d) 7
21. 21.What is the units digit of 9^42? (a) 2 (b) 9 (c) 1 (d) 0
22. 22.What is the smallest number you must add to 296 to make it divisible by 9? (a) 3 (b) 2 (c) 1 (d) 8
23. 23.What is the units digit of 2^24? (a) 4 (b) 6 (c) 2 (d) 8
24. 24.Which of these numbers is divisible by 11? (a) 1829 (b) 1812 (c) 1815 (d) 1816
25. 25.What is the remainder when 259 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
26. 26.Which of these numbers is divisible by 6? (a) 3066 (b) 3061 (c) 3068 (d) 3073
27. 27.What is the remainder when 47 is divided by 3? (a) 0 (b) 2 (c) 1
28. 28.What is the smallest number you must add to 584 to make it divisible by 9? (a) 8 (b) 1 (c) 3 (d) 2
29. 29.What is the units digit of 3^19? (a) 1 (b) 7 (c) 3 (d) 9
30. 30.Which of these numbers is divisible by 11? (a) 1143 (b) 1149 (c) 1133 (d) 1124
31. 31.What is the units digit of 2^55? (a) 6 (b) 4 (c) 8 (d) 2
32. 32.What is the units digit of 8^14? (a) 8 (b) 6 (c) 4 (d) 2
33. 33.Which of these numbers is divisible by 4? (a) 3110 (b) 3103 (c) 3105 (d) 3104
34. 34.Which of these numbers is divisible by 8? (a) 870 (b) 877 (c) 872 (d) 885
35. 35.What is the units digit of 8^57? (a) 6 (b) 2 (c) 8 (d) 4
36. 36.What is the smallest number you must add to 654 to make it divisible by 11? (a) 7 (b) 5 (c) 8 (d) 6
37. 37.What is the remainder when 791 is divided by 10? (a) 9 (b) 0 (c) 2 (d) 1
38. 38.What is the smallest number you must add to 429 to make it divisible by 8? (a) 3 (b) 5 (c) 4 (d) 2
39. 39.What is the smallest number you must add to 562 to make it divisible by 12? (a) 10 (b) 1 (c) 2 (d) 3
40. 40.What is the smallest number you must add to 221 to make it divisible by 12? (a) 6 (b) 7 (c) 5 (d) 8

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