Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 438 · 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 4^54? (a) 4 (b) 5 (c) 7 (d) 6
2. 2.What is the units digit of 8^45? (a) 4 (b) 2 (c) 6 (d) 8
3. 3.What is the units digit of 4^10? (a) 4 (b) 6 (c) 5 (d) 7
4. 4.What is the smallest number you must add to 94 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 1
5. 5.What is the units digit of 8^10? (a) 4 (b) 6 (c) 2 (d) 8
6. 6.What is the units digit of 7^27? (a) 9 (b) 7 (c) 1 (d) 3
7. 7.Which of these numbers is divisible by 4? (a) 2744 (b) 2747 (c) 2751 (d) 2743
8. 8.What is the smallest number you must add to 885 to make it divisible by 12? (a) 9 (b) 4 (c) 2 (d) 3
9. 9.What is the smallest number you must add to 444 to make it divisible by 11? (a) 4 (b) 7 (c) 6 (d) 8
10. 10.What is the remainder when 610 is divided by 8? (a) 3 (b) 2 (c) 1 (d) 6
11. 11.What is the units digit of 9^44? (a) 2 (b) 9 (c) 1 (d) 0
12. 12.What is the smallest number you must add to 551 to make it divisible by 9? (a) 7 (b) 2 (c) 8 (d) 6
13. 13.What is the smallest number you must add to 561 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 1
14. 14.What is the units digit of 7^35? (a) 7 (b) 1 (c) 9 (d) 3
15. 15.Which of these numbers is divisible by 6? (a) 5196 (b) 5199 (c) 5194 (d) 5207
16. 16.What is the smallest number you must add to 84 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
17. 17.Which of these numbers is divisible by 8? (a) 5877 (b) 5872 (c) 5870 (d) 5883
18. 18.What is the remainder when 205 is divided by 3? (a) 1 (b) 2 (c) 0
19. 19.Which of these numbers is divisible by 4? (a) 2136 (b) 2139 (c) 2142 (d) 2135
20. 20.What is the remainder when 550 is divided by 7? (a) 6 (b) 5 (c) 3 (d) 4
21. 21.What is the remainder when 163 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
22. 22.Which of these numbers is divisible by 3? (a) 249 (b) 248 (c) 254 (d) 250
23. 23.What is the units digit of 9^51? (a) 1 (b) 8 (c) 7 (d) 9
24. 24.What is the units digit of 7^46? (a) 1 (b) 7 (c) 9 (d) 3
25. 25.What is the remainder when 276 is divided by 5? (a) 0 (b) 1 (c) 4 (d) 2
26. 26.What is the smallest number you must add to 233 to make it divisible by 8? (a) 7 (b) 1 (c) 6 (d) 5
27. 27.Which of these numbers is divisible by 8? (a) 2371 (b) 2363 (c) 2377 (d) 2368
28. 28.Which of these numbers is divisible by 8? (a) 1171 (b) 1176 (c) 1189 (d) 1177
29. 29.What is the units digit of 7^10? (a) 7 (b) 3 (c) 9 (d) 1
30. 30.What is the units digit of 7^21? (a) 9 (b) 7 (c) 3 (d) 1
31. 31.What is the smallest number you must add to 246 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
32. 32.Which of these numbers is divisible by 9? (a) 1469 (b) 1466 (c) 1483 (d) 1467
33. 33.What is the smallest number you must add to 324 to make it divisible by 10? (a) 4 (b) 5 (c) 7 (d) 6
34. 34.What is the smallest number you must add to 218 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
35. 35.Which of these numbers is divisible by 4? (a) 1265 (b) 1269 (c) 1263 (d) 1264
36. 36.Which of these numbers is divisible by 9? (a) 2619 (b) 2624 (c) 2614 (d) 2629
37. 37.Which of these numbers is divisible by 6? (a) 1369 (b) 1381 (c) 1374 (d) 1378
38. 38.What is the remainder when 209 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
39. 39.What is the units digit of 3^39? (a) 9 (b) 3 (c) 7 (d) 1
40. 40.Which of these numbers is divisible by 6? (a) 3371 (b) 3372 (c) 3381 (d) 3376

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