Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 172 · 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 4^25? (a) 6 (b) 3 (c) 5 (d) 4
2. 2.What is the smallest number you must add to 145 to make it divisible by 11? (a) 8 (b) 10 (c) 9 (d) 2
3. 3.What is the remainder when 557 is divided by 9? (a) 7 (b) 6 (c) 1 (d) 8
4. 4.Which of these numbers is divisible by 3? (a) 1261 (b) 1264 (c) 1267 (d) 1263
5. 5.What is the units digit of 2^19? (a) 6 (b) 4 (c) 8 (d) 2
6. 6.What is the units digit of 2^48? (a) 8 (b) 6 (c) 4 (d) 2
7. 7.What is the remainder when 591 is divided by 11? (a) 9 (b) 7 (c) 8 (d) 3
8. 8.Which of these numbers is divisible by 9? (a) 3118 (b) 3124 (c) 3136 (d) 3123
9. 9.What is the units digit of 7^54? (a) 3 (b) 7 (c) 9 (d) 1
10. 10.Which of these numbers is divisible by 4? (a) 573 (b) 583 (c) 576 (d) 579
11. 11.What is the smallest number you must add to 175 to make it divisible by 3? (a) 1 (b) 2
12. 12.Which of these numbers is divisible by 3? (a) 2279 (b) 2276 (c) 2277 (d) 2282
13. 13.What is the smallest number you must add to 545 to make it divisible by 7? (a) 1 (b) 2 (c) 6 (d) 3
14. 14.Which of these numbers is divisible by 9? (a) 6282 (b) 6276 (c) 6298 (d) 6286
15. 15.What is the remainder when 474 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 2
16. 16.What is the units digit of 9^50? (a) 1 (b) 0 (c) 9 (d) 2
17. 17.Which of these numbers is divisible by 11? (a) 3466 (b) 3462 (c) 3454 (d) 3449
18. 18.What is the smallest number you must add to 287 to make it divisible by 12? (a) 2 (b) 11 (c) 3 (d) 1
19. 19.Which of these numbers is divisible by 3? (a) 1840 (b) 1843 (c) 1842 (d) 1847
20. 20.Which of these numbers is divisible by 6? (a) 5293 (b) 5298 (c) 5300 (d) 5305
21. 21.What is the units digit of 8^40? (a) 8 (b) 4 (c) 6 (d) 2
22. 22.What is the units digit of 8^20? (a) 6 (b) 2 (c) 4 (d) 8
23. 23.What is the remainder when 471 is divided by 9? (a) 3 (b) 4 (c) 6 (d) 2
24. 24.What is the units digit of 3^41? (a) 7 (b) 1 (c) 3 (d) 9
25. 25.What is the units digit of 2^18? (a) 8 (b) 2 (c) 6 (d) 4
26. 26.What is the smallest number you must add to 549 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 2
27. 27.What is the smallest number you must add to 244 to make it divisible by 9? (a) 6 (b) 1 (c) 7 (d) 8
28. 28.What is the remainder when 47 is divided by 3? (a) 2 (b) 0 (c) 1
29. 29.What is the remainder when 298 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 5
30. 30.What is the units digit of 3^18? (a) 9 (b) 7 (c) 3 (d) 1
31. 31.Which of these numbers is divisible by 11? (a) 2893 (b) 2896 (c) 2892 (d) 2910
32. 32.What is the smallest number you must add to 109 to make it divisible by 9? (a) 6 (b) 8 (c) 1 (d) 7
33. 33.What is the smallest number you must add to 716 to make it divisible by 12? (a) 4 (b) 8 (c) 3 (d) 5
34. 34.What is the units digit of 8^21? (a) 2 (b) 4 (c) 6 (d) 8
35. 35.What is the remainder when 49 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
36. 36.What is the smallest number you must add to 419 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 5
37. 37.What is the units digit of 4^45? (a) 4 (b) 3 (c) 5 (d) 6
38. 38.What is the remainder when 469 is divided by 12? (a) 1 (b) 11 (c) 0 (d) 2
39. 39.Which of these numbers is divisible by 9? (a) 1134 (b) 1145 (c) 1135 (d) 1132
40. 40.What is the units digit of 2^45? (a) 2 (b) 4 (c) 6 (d) 8

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