Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 176 · 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 2^14? (a) 4 (b) 6 (c) 2 (d) 8
2. 2.What is the units digit of 2^55? (a) 6 (b) 2 (c) 8 (d) 4
3. 3.What is the smallest number you must add to 164 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 2
4. 4.What is the smallest number you must add to 570 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
5. 5.Which of these numbers is divisible by 4? (a) 2710 (b) 2713 (c) 2712 (d) 2717
6. 6.What is the smallest number you must add to 199 to make it divisible by 7? (a) 3 (b) 4 (c) 5 (d) 6
7. 7.What is the units digit of 3^26? (a) 7 (b) 9 (c) 3 (d) 1
8. 8.What is the units digit of 7^42? (a) 3 (b) 7 (c) 1 (d) 9
9. 9.Which of these numbers is divisible by 3? (a) 236 (b) 238 (c) 241 (d) 237
10. 10.Which of these numbers is divisible by 9? (a) 1182 (b) 1195 (c) 1188 (d) 1198
11. 11.What is the smallest number you must add to 34 to make it divisible by 3? (a) 1 (b) 2
12. 12.What is the smallest number you must add to 690 to make it divisible by 12? (a) 8 (b) 5 (c) 7 (d) 6
13. 13.Which of these numbers is divisible by 6? (a) 1438 (b) 1443 (c) 1434 (d) 1433
14. 14.What is the smallest number you must add to 224 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
15. 15.What is the remainder when 129 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
16. 16.Which of these numbers is divisible by 3? (a) 1592 (b) 1594 (c) 1593 (d) 1597
17. 17.What is the smallest number you must add to 644 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 6
18. 18.What is the smallest number you must add to 223 to make it divisible by 3? (a) 2 (b) 1
19. 19.Which of these numbers is divisible by 4? (a) 2303 (b) 2305 (c) 2300 (d) 2298
20. 20.What is the remainder when 235 is divided by 3? (a) 2 (b) 1 (c) 0
21. 21.Which of these numbers is divisible by 3? (a) 739 (b) 737 (c) 738 (d) 743
22. 22.Which of these numbers is divisible by 9? (a) 8070 (b) 8052 (c) 8058 (d) 8055
23. 23.Which of these numbers is divisible by 4? (a) 1673 (b) 1681 (c) 1678 (d) 1676
24. 24.What is the units digit of 9^20? (a) 9 (b) 1 (c) 2 (d) 0
25. 25.What is the smallest number you must add to 322 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
26. 26.What is the units digit of 4^11? (a) 4 (b) 3 (c) 6 (d) 5
27. 27.What is the smallest number you must add to 208 to make it divisible by 6? (a) 1 (b) 3 (c) 4 (d) 2
28. 28.Which of these numbers is divisible by 9? (a) 1316 (b) 1311 (c) 1303 (d) 1305
29. 29.Which of these numbers is divisible by 11? (a) 3354 (b) 3339 (c) 3357 (d) 3344
30. 30.What is the remainder when 307 is divided by 10? (a) 6 (b) 8 (c) 7 (d) 3
31. 31.Which of these numbers is divisible by 4? (a) 2467 (b) 2468 (c) 2474 (d) 2470
32. 32.Which of these numbers is divisible by 6? (a) 5136 (b) 5147 (c) 5137 (d) 5134
33. 33.What is the remainder when 776 is divided by 10? (a) 5 (b) 6 (c) 7 (d) 4
34. 34.What is the smallest number you must add to 121 to make it divisible by 8? (a) 7 (b) 6 (c) 5 (d) 1
35. 35.Which of these numbers is divisible by 11? (a) 9448 (b) 9428 (c) 9438 (d) 9454
36. 36.Which of these numbers is divisible by 6? (a) 321 (b) 316 (c) 312 (d) 308
37. 37.What is the units digit of 7^14? (a) 9 (b) 3 (c) 7 (d) 1
38. 38.What is the remainder when 138 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
39. 39.What is the units digit of 3^15? (a) 3 (b) 1 (c) 9 (d) 7
40. 40.What is the smallest number you must add to 316 to make it divisible by 11? (a) 4 (b) 8 (c) 3 (d) 2

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