Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 983 · 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.Which of these numbers is divisible by 9? (a) 4563 (b) 4567 (c) 4578 (d) 4560
2. 2.What is the smallest number you must add to 292 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
3. 3.What is the smallest number you must add to 93 to make it divisible by 9? (a) 6 (b) 7 (c) 3 (d) 5
4. 4.What is the units digit of 7^12? (a) 1 (b) 9 (c) 3 (d) 7
5. 5.What is the smallest number you must add to 629 to make it divisible by 9? (a) 2 (b) 3 (c) 8 (d) 1
6. 6.What is the remainder when 421 is divided by 11? (a) 8 (b) 4 (c) 2 (d) 3
7. 7.Which of these numbers is divisible by 3? (a) 2461 (b) 2463 (c) 2465 (d) 2467
8. 8.Which of these numbers is divisible by 8? (a) 2272 (b) 2278 (c) 2268 (d) 2282
9. 9.What is the units digit of 8^43? (a) 4 (b) 8 (c) 6 (d) 2
10. 10.What is the units digit of 3^39? (a) 1 (b) 3 (c) 9 (d) 7
11. 11.What is the units digit of 3^44? (a) 9 (b) 3 (c) 7 (d) 1
12. 12.Which of these numbers is divisible by 8? (a) 1007 (b) 1009 (c) 1000 (d) 998
13. 13.What is the smallest number you must add to 293 to make it divisible by 12? (a) 8 (b) 5 (c) 7 (d) 6
14. 14.What is the units digit of 3^14? (a) 7 (b) 3 (c) 9 (d) 1
15. 15.What is the units digit of 4^51? (a) 6 (b) 3 (c) 5 (d) 4
16. 16.What is the units digit of 4^58? (a) 7 (b) 6 (c) 5 (d) 4
17. 17.What is the smallest number you must add to 485 to make it divisible by 8? (a) 3 (b) 5 (c) 2 (d) 4
18. 18.What is the units digit of 9^15? (a) 7 (b) 8 (c) 1 (d) 9
19. 19.What is the remainder when 732 is divided by 11? (a) 5 (b) 8 (c) 7 (d) 6
20. 20.Which of these numbers is divisible by 11? (a) 5951 (b) 5953 (c) 5963 (d) 5947
21. 21.What is the remainder when 617 is divided by 12? (a) 7 (b) 4 (c) 6 (d) 5
22. 22.What is the remainder when 679 is divided by 10? (a) 8 (b) 7 (c) 9 (d) 1
23. 23.What is the remainder when 729 is divided by 10? (a) 1 (b) 8 (c) 9 (d) 7
24. 24.What is the remainder when 172 is divided by 10? (a) 8 (b) 1 (c) 2 (d) 3
25. 25.Which of these numbers is divisible by 8? (a) 2805 (b) 2808 (c) 2820 (d) 2811
26. 26.Which of these numbers is divisible by 6? (a) 4408 (b) 4396 (c) 4401 (d) 4398
27. 27.What is the units digit of 4^16? (a) 6 (b) 5 (c) 7 (d) 4
28. 28.Which of these numbers is divisible by 4? (a) 3366 (b) 3368 (c) 3370 (d) 3373
29. 29.Which of these numbers is divisible by 9? (a) 2418 (b) 2412 (c) 2424 (d) 2408
30. 30.What is the smallest number you must add to 245 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 5
31. 31.What is the smallest number you must add to 202 to make it divisible by 7? (a) 2 (b) 1 (c) 6 (d) 3
32. 32.Which of these numbers is divisible by 3? (a) 2700 (b) 2699 (c) 2702 (d) 2705
33. 33.What is the units digit of 7^17? (a) 9 (b) 3 (c) 1 (d) 7
34. 34.What is the smallest number you must add to 515 to make it divisible by 12? (a) 3 (b) 11 (c) 2 (d) 1
35. 35.Which of these numbers is divisible by 4? (a) 1270 (b) 1266 (c) 1274 (d) 1268
36. 36.What is the remainder when 818 is divided by 12? (a) 2 (b) 3 (c) 1 (d) 10
37. 37.Which of these numbers is divisible by 3? (a) 304 (b) 310 (c) 306 (d) 308
38. 38.Which of these numbers is divisible by 8? (a) 3189 (b) 3180 (c) 3174 (d) 3176
39. 39.What is the smallest number you must add to 75 to make it divisible by 4? (a) 2 (b) 1 (c) 3
40. 40.What is the smallest number you must add to 195 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5

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