Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 296 · 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 148 to make it divisible by 7? (a) 6 (b) 4 (c) 1 (d) 5
2. 2.What is the units digit of 9^35? (a) 8 (b) 9 (c) 7 (d) 1
3. 3.What is the units digit of 3^39? (a) 9 (b) 1 (c) 7 (d) 3
4. 4.What is the remainder when 721 is divided by 10? (a) 1 (b) 9 (c) 0 (d) 2
5. 5.What is the remainder when 380 is divided by 12? (a) 9 (b) 8 (c) 7 (d) 4
6. 6.What is the smallest number you must add to 121 to make it divisible by 4? (a) 1 (b) 3 (c) 2
7. 7.Which of these numbers is divisible by 3? (a) 1326 (b) 1325 (c) 1330 (d) 1327
8. 8.What is the units digit of 2^27? (a) 6 (b) 8 (c) 4 (d) 2
9. 9.Which of these numbers is divisible by 8? (a) 1478 (b) 1480 (c) 1492 (d) 1485
10. 10.Which of these numbers is divisible by 3? (a) 289 (b) 292 (c) 286 (d) 288
11. 11.What is the smallest number you must add to 406 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 1
12. 12.Which of these numbers is divisible by 9? (a) 2983 (b) 2975 (c) 2979 (d) 2989
13. 13.What is the units digit of 9^32? (a) 1 (b) 9 (c) 2 (d) 0
14. 14.Which of these numbers is divisible by 3? (a) 1609 (b) 1612 (c) 1611 (d) 1615
15. 15.What is the smallest number you must add to 167 to make it divisible by 11? (a) 8 (b) 9 (c) 2 (d) 10
16. 16.Which of these numbers is divisible by 8? (a) 5522 (b) 5515 (c) 5529 (d) 5520
17. 17.What is the units digit of 7^60? (a) 7 (b) 3 (c) 9 (d) 1
18. 18.What is the smallest number you must add to 478 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 2
19. 19.What is the smallest number you must add to 412 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 6
20. 20.Which of these numbers is divisible by 8? (a) 2251 (b) 2240 (c) 2236 (d) 2242
21. 21.What is the units digit of 7^44? (a) 7 (b) 1 (c) 9 (d) 3
22. 22.What is the remainder when 247 is divided by 9? (a) 3 (b) 5 (c) 4 (d) 6
23. 23.What is the units digit of 8^48? (a) 8 (b) 6 (c) 2 (d) 4
24. 24.Which of these numbers is divisible by 3? (a) 2622 (b) 2626 (c) 2621 (d) 2623
25. 25.What is the smallest number you must add to 179 to make it divisible by 3? (a) 2 (b) 1
26. 26.What is the smallest number you must add to 446 to make it divisible by 12? (a) 11 (b) 9 (c) 10 (d) 2
27. 27.What is the units digit of 4^18? (a) 6 (b) 4 (c) 5 (d) 7
28. 28.What is the units digit of 9^14? (a) 1 (b) 9 (c) 2 (d) 0
29. 29.Which of these numbers is divisible by 8? (a) 4991 (b) 4982 (c) 4998 (d) 4984
30. 30.What is the smallest number you must add to 237 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
31. 31.What is the units digit of 7^27? (a) 3 (b) 1 (c) 9 (d) 7
32. 32.What is the units digit of 3^16? (a) 3 (b) 1 (c) 7 (d) 9
33. 33.What is the smallest number you must add to 746 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 9
34. 34.Which of these numbers is divisible by 4? (a) 1135 (b) 1131 (c) 1128 (d) 1125
35. 35.What is the smallest number you must add to 542 to make it divisible by 12? (a) 9 (b) 11 (c) 10 (d) 2
36. 36.Which of these numbers is divisible by 9? (a) 5049 (b) 5062 (c) 5043 (d) 5056
37. 37.Which of these numbers is divisible by 3? (a) 292 (b) 298 (c) 294 (d) 295
38. 38.Which of these numbers is divisible by 6? (a) 2538 (b) 2540 (c) 2536 (d) 2548
39. 39.What is the units digit of 7^21? (a) 9 (b) 3 (c) 7 (d) 1
40. 40.What is the remainder when 306 is divided by 10? (a) 4 (b) 7 (c) 5 (d) 6

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