Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 155 · 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 4^21? (a) 5 (b) 6 (c) 3 (d) 4
2. 2.What is the smallest number you must add to 266 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
3. 3.What is the units digit of 4^28? (a) 4 (b) 5 (c) 7 (d) 6
4. 4.Which of these numbers is divisible by 11? (a) 1092 (b) 1109 (c) 1082 (d) 1089
5. 5.What is the smallest number you must add to 161 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
6. 6.Which of these numbers is divisible by 6? (a) 2688 (b) 2690 (c) 2695 (d) 2683
7. 7.What is the smallest number you must add to 76 to make it divisible by 3? (a) 2 (b) 1
8. 8.What is the remainder when 418 is divided by 8? (a) 1 (b) 3 (c) 2 (d) 6
9. 9.What is the remainder when 164 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
10. 10.What is the units digit of 3^13? (a) 1 (b) 3 (c) 7 (d) 9
11. 11.What is the units digit of 2^29? (a) 8 (b) 4 (c) 2 (d) 6
12. 12.Which of these numbers is divisible by 11? (a) 451 (b) 450 (c) 463 (d) 458
13. 13.What is the units digit of 2^49? (a) 8 (b) 2 (c) 6 (d) 4
14. 14.What is the remainder when 398 is divided by 9? (a) 7 (b) 1 (c) 2 (d) 3
15. 15.What is the smallest number you must add to 346 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
16. 16.What is the smallest number you must add to 92 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
17. 17.What is the remainder when 68 is divided by 3? (a) 2 (b) 1 (c) 0
18. 18.What is the smallest number you must add to 746 to make it divisible by 11? (a) 9 (b) 1 (c) 2 (d) 3
19. 19.What is the units digit of 3^16? (a) 9 (b) 3 (c) 7 (d) 1
20. 20.What is the remainder when 259 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
21. 21.What is the smallest number you must add to 201 to make it divisible by 10? (a) 1 (b) 8 (c) 9 (d) 7
22. 22.What is the smallest number you must add to 89 to make it divisible by 3? (a) 2 (b) 1
23. 23.What is the smallest number you must add to 304 to make it divisible by 7? (a) 5 (b) 6 (c) 3 (d) 4
24. 24.What is the units digit of 9^58? (a) 9 (b) 0 (c) 2 (d) 1
25. 25.Which of these numbers is divisible by 9? (a) 1334 (b) 1320 (c) 1330 (d) 1323
26. 26.Which of these numbers is divisible by 3? (a) 2203 (b) 2205 (c) 2207 (d) 2209
27. 27.Which of these numbers is divisible by 3? (a) 1202 (b) 1197 (c) 1199 (d) 1195
28. 28.What is the smallest number you must add to 797 to make it divisible by 11? (a) 5 (b) 6 (c) 8 (d) 7
29. 29.Which of these numbers is divisible by 4? (a) 1237 (b) 1232 (c) 1230 (d) 1235
30. 30.Which of these numbers is divisible by 6? (a) 1657 (b) 1671 (c) 1662 (d) 1665
31. 31.Which of these numbers is divisible by 3? (a) 2388 (b) 2389 (c) 2392 (d) 2387
32. 32.What is the units digit of 8^36? (a) 8 (b) 2 (c) 4 (d) 6
33. 33.Which of these numbers is divisible by 6? (a) 2752 (b) 2741 (c) 2743 (d) 2742
34. 34.What is the units digit of 7^32? (a) 9 (b) 3 (c) 1 (d) 7
35. 35.What is the remainder when 447 is divided by 6? (a) 5 (b) 2 (c) 3 (d) 4
36. 36.What is the units digit of 8^34? (a) 2 (b) 8 (c) 6 (d) 4
37. 37.What is the remainder when 199 is divided by 7? (a) 4 (b) 3 (c) 5 (d) 2
38. 38.What is the units digit of 7^36? (a) 3 (b) 9 (c) 7 (d) 1
39. 39.What is the smallest number you must add to 98 to make it divisible by 9? (a) 2 (b) 8 (c) 3 (d) 1
40. 40.What is the remainder when 77 is divided by 3? (a) 1 (b) 0 (c) 2

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