Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 957 · 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 7^19? (a) 3 (b) 7 (c) 9 (d) 1
2. 2.Which of these numbers is divisible by 8? (a) 5012 (b) 5022 (c) 5008 (d) 5004
3. 3.Which of these numbers is divisible by 4? (a) 3266 (b) 3271 (c) 3274 (d) 3268
4. 4.What is the units digit of 9^25? (a) 8 (b) 1 (c) 7 (d) 9
5. 5.Which of these numbers is divisible by 4? (a) 525 (b) 520 (c) 522 (d) 517
6. 6.What is the remainder when 128 is divided by 3? (a) 2 (b) 0 (c) 1
7. 7.Which of these numbers is divisible by 4? (a) 1151 (b) 1142 (c) 1146 (d) 1144
8. 8.Which of these numbers is divisible by 8? (a) 888 (b) 893 (c) 881 (d) 902
9. 9.What is the units digit of 2^51? (a) 4 (b) 2 (c) 6 (d) 8
10. 10.Which of these numbers is divisible by 9? (a) 7535 (b) 7542 (c) 7558 (d) 7543
11. 11.What is the smallest number you must add to 47 to make it divisible by 3? (a) 1 (b) 2
12. 12.What is the units digit of 9^59? (a) 7 (b) 1 (c) 9 (d) 8
13. 13.What is the smallest number you must add to 560 to make it divisible by 11? (a) 10 (b) 2 (c) 1 (d) 3
14. 14.Which of these numbers is divisible by 6? (a) 1255 (b) 1262 (c) 1252 (d) 1254
15. 15.What is the remainder when 482 is divided by 8? (a) 2 (b) 3 (c) 1 (d) 6
16. 16.Which of these numbers is divisible by 8? (a) 4715 (b) 4735 (c) 4720 (d) 4722
17. 17.Which of these numbers is divisible by 9? (a) 985 (b) 996 (c) 980 (d) 981
18. 18.What is the smallest number you must add to 34 to make it divisible by 3? (a) 1 (b) 2
19. 19.Which of these numbers is divisible by 4? (a) 3337 (b) 3331 (c) 3332 (d) 3335
20. 20.What is the remainder when 336 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
21. 21.What is the remainder when 387 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
22. 22.What is the units digit of 7^36? (a) 7 (b) 1 (c) 3 (d) 9
23. 23.Which of these numbers is divisible by 4? (a) 2842 (b) 2839 (c) 2835 (d) 2836
24. 24.What is the units digit of 2^28? (a) 8 (b) 2 (c) 6 (d) 4
25. 25.What is the units digit of 8^48? (a) 2 (b) 8 (c) 4 (d) 6
26. 26.What is the units digit of 8^38? (a) 2 (b) 4 (c) 8 (d) 6
27. 27.What is the units digit of 8^58? (a) 4 (b) 6 (c) 2 (d) 8
28. 28.What is the smallest number you must add to 161 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
29. 29.Which of these numbers is divisible by 6? (a) 2692 (b) 2684 (c) 2688 (d) 2695
30. 30.What is the remainder when 68 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3
31. 31.What is the remainder when 652 is divided by 10? (a) 2 (b) 1 (c) 8 (d) 3
32. 32.Which of these numbers is divisible by 3? (a) 1258 (b) 1257 (c) 1261 (d) 1256
33. 33.What is the remainder when 275 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 7
34. 34.Which of these numbers is divisible by 11? (a) 690 (b) 673 (c) 696 (d) 682
35. 35.Which of these numbers is divisible by 8? (a) 6168 (b) 6173 (c) 6163 (d) 6179
36. 36.What is the units digit of 2^52? (a) 2 (b) 6 (c) 8 (d) 4
37. 37.What is the units digit of 2^27? (a) 4 (b) 6 (c) 8 (d) 2
38. 38.Which of these numbers is divisible by 3? (a) 574 (b) 581 (c) 578 (d) 576
39. 39.What is the units digit of 7^25? (a) 1 (b) 9 (c) 7 (d) 3
40. 40.What is the smallest number you must add to 259 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4

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