Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 124 · 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 smallest number you must add to 449 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 5
2. 2.What is the smallest number you must add to 582 to make it divisible by 12? (a) 5 (b) 7 (c) 6 (d) 8
3. 3.Which of these numbers is divisible by 3? (a) 935 (b) 928 (c) 930 (d) 932
4. 4.What is the units digit of 7^48? (a) 3 (b) 1 (c) 7 (d) 9
5. 5.What is the smallest number you must add to 166 to make it divisible by 11? (a) 8 (b) 9 (c) 10 (d) 1
6. 6.Which of these numbers is divisible by 11? (a) 8952 (b) 8937 (c) 8932 (d) 8927
7. 7.What is the units digit of 7^59? (a) 7 (b) 9 (c) 1 (d) 3
8. 8.What is the units digit of 3^32? (a) 9 (b) 7 (c) 3 (d) 1
9. 9.What is the units digit of 8^46? (a) 8 (b) 4 (c) 6 (d) 2
10. 10.Which of these numbers is divisible by 4? (a) 543 (b) 544 (c) 546 (d) 550
11. 11.What is the remainder when 752 is divided by 11? (a) 5 (b) 3 (c) 7 (d) 4
12. 12.Which of these numbers is divisible by 3? (a) 1369 (b) 1367 (c) 1365 (d) 1363
13. 13.What is the units digit of 8^31? (a) 8 (b) 6 (c) 2 (d) 4
14. 14.Which of these numbers is divisible by 3? (a) 266 (b) 264 (c) 268 (d) 262
15. 15.What is the units digit of 3^39? (a) 9 (b) 3 (c) 1 (d) 7
16. 16.What is the units digit of 7^24? (a) 3 (b) 1 (c) 7 (d) 9
17. 17.Which of these numbers is divisible by 6? (a) 1468 (b) 1473 (c) 1477 (d) 1470
18. 18.Which of these numbers is divisible by 3? (a) 2599 (b) 2595 (c) 2593 (d) 2597
19. 19.What is the units digit of 8^22? (a) 2 (b) 8 (c) 6 (d) 4
20. 20.What is the smallest number you must add to 78 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
21. 21.What is the remainder when 293 is divided by 6? (a) 1 (b) 5 (c) 4 (d) 3
22. 22.Which of these numbers is divisible by 4? (a) 1318 (b) 1320 (c) 1323 (d) 1325
23. 23.What is the remainder when 273 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
24. 24.What is the units digit of 3^22? (a) 9 (b) 1 (c) 7 (d) 3
25. 25.What is the remainder when 139 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
26. 26.Which of these numbers is divisible by 11? (a) 3170 (b) 3168 (c) 3163 (d) 3186
27. 27.Which of these numbers is divisible by 11? (a) 5576 (b) 5559 (c) 5579 (d) 5566
28. 28.What is the remainder when 75 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
29. 29.What is the smallest number you must add to 193 to make it divisible by 12? (a) 1 (b) 10 (c) 11 (d) 9
30. 30.What is the smallest number you must add to 701 to make it divisible by 11? (a) 4 (b) 2 (c) 3 (d) 8
31. 31.What is the smallest number you must add to 257 to make it divisible by 10? (a) 7 (b) 4 (c) 2 (d) 3
32. 32.What is the smallest number you must add to 83 to make it divisible by 3? (a) 1 (b) 2
33. 33.Which of these numbers is divisible by 9? (a) 2504 (b) 2526 (c) 2517 (d) 2511
34. 34.What is the smallest number you must add to 138 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
35. 35.What is the remainder when 604 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 3
36. 36.What is the units digit of 4^36? (a) 7 (b) 5 (c) 4 (d) 6
37. 37.What is the units digit of 7^17? (a) 1 (b) 7 (c) 3 (d) 9
38. 38.What is the remainder when 469 is divided by 10? (a) 8 (b) 9 (c) 7 (d) 1
39. 39.What is the smallest number you must add to 172 to make it divisible by 6? (a) 1 (b) 2 (c) 3 (d) 4
40. 40.What is the smallest number you must add to 632 to make it divisible by 10? (a) 7 (b) 2 (c) 9 (d) 8

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