Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 122 · 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 remainder when 75 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 5
2. 2.What is the remainder when 612 is divided by 10? (a) 8 (b) 1 (c) 2 (d) 3
3. 3.What is the units digit of 8^20? (a) 4 (b) 8 (c) 2 (d) 6
4. 4.What is the units digit of 3^42? (a) 9 (b) 7 (c) 3 (d) 1
5. 5.What is the units digit of 4^45? (a) 6 (b) 5 (c) 3 (d) 4
6. 6.Which of these numbers is divisible by 4? (a) 1595 (b) 1597 (c) 1592 (d) 1589
7. 7.Which of these numbers is divisible by 11? (a) 3252 (b) 3271 (c) 3256 (d) 3263
8. 8.Which of these numbers is divisible by 6? (a) 4355 (b) 4356 (c) 4366 (d) 4360
9. 9.What is the smallest number you must add to 610 to make it divisible by 9? (a) 1 (b) 3 (c) 2 (d) 7
10. 10.What is the smallest number you must add to 359 to make it divisible by 10? (a) 3 (b) 9 (c) 1 (d) 2
11. 11.What is the smallest number you must add to 310 to make it divisible by 9? (a) 6 (b) 7 (c) 5 (d) 4
12. 12.Which of these numbers is divisible by 3? (a) 2352 (b) 2354 (c) 2356 (d) 2350
13. 13.What is the remainder when 164 is divided by 9? (a) 3 (b) 2 (c) 7 (d) 1
14. 14.What is the units digit of 9^49? (a) 8 (b) 9 (c) 1 (d) 7
15. 15.What is the units digit of 4^56? (a) 6 (b) 4 (c) 5 (d) 7
16. 16.What is the smallest number you must add to 64 to make it divisible by 3? (a) 1 (b) 2
17. 17.What is the units digit of 9^31? (a) 1 (b) 9 (c) 7 (d) 8
18. 18.Which of these numbers is divisible by 9? (a) 2981 (b) 2970 (c) 2975 (d) 2966
19. 19.What is the smallest number you must add to 427 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 1
20. 20.What is the units digit of 4^14? (a) 7 (b) 5 (c) 6 (d) 4
21. 21.Which of these numbers is divisible by 9? (a) 2286 (b) 2278 (c) 2288 (d) 2300
22. 22.What is the remainder when 623 is divided by 8? (a) 5 (b) 1 (c) 7 (d) 6
23. 23.What is the smallest number you must add to 487 to make it divisible by 9? (a) 1 (b) 6 (c) 7 (d) 8
24. 24.What is the remainder when 98 is divided by 3? (a) 1 (b) 2 (c) 0
25. 25.What is the remainder when 244 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
26. 26.What is the units digit of 3^13? (a) 7 (b) 3 (c) 1 (d) 9
27. 27.What is the remainder when 119 is divided by 11? (a) 2 (b) 8 (c) 10 (d) 9
28. 28.Which of these numbers is divisible by 3? (a) 483 (b) 487 (c) 485 (d) 482
29. 29.Which of these numbers is divisible by 3? (a) 1257 (b) 1261 (c) 1259 (d) 1255
30. 30.Which of these numbers is divisible by 11? (a) 1949 (b) 1945 (c) 1959 (d) 1947
31. 31.Which of these numbers is divisible by 8? (a) 6103 (b) 6094 (c) 6096 (d) 6111
32. 32.What is the smallest number you must add to 295 to make it divisible by 8? (a) 2 (b) 1 (c) 3 (d) 7
33. 33.Which of these numbers is divisible by 6? (a) 4877 (b) 4872 (c) 4880 (d) 4871
34. 34.What is the smallest number you must add to 56 to make it divisible by 3? (a) 1 (b) 2
35. 35.What is the smallest number you must add to 323 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 1
36. 36.Which of these numbers is divisible by 4? (a) 2906 (b) 2904 (c) 2902 (d) 2911
37. 37.What is the remainder when 582 is divided by 11? (a) 9 (b) 1 (c) 8 (d) 10
38. 38.What is the units digit of 9^48? (a) 9 (b) 1 (c) 2 (d) 0
39. 39.What is the units digit of 7^60? (a) 9 (b) 3 (c) 1 (d) 7
40. 40.What is the smallest number you must add to 323 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3

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