Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 430 · 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 81 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
2. 2.What is the remainder when 73 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
3. 3.What is the smallest number you must add to 130 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the units digit of 3^26? (a) 1 (b) 7 (c) 9 (d) 3
5. 5.What is the smallest number you must add to 107 to make it divisible by 3? (a) 1 (b) 2
6. 6.What is the smallest number you must add to 40 to make it divisible by 3? (a) 1 (b) 2
7. 7.Which of these numbers is divisible by 3? (a) 2359 (b) 2357 (c) 2354 (d) 2355
8. 8.Which of these numbers is divisible by 9? (a) 5495 (b) 5489 (c) 5490 (d) 5500
9. 9.What is the units digit of 2^18? (a) 4 (b) 2 (c) 8 (d) 6
10. 10.What is the units digit of 8^14? (a) 4 (b) 6 (c) 2 (d) 8
11. 11.Which of these numbers is divisible by 8? (a) 5965 (b) 5968 (c) 5977 (d) 5971
12. 12.Which of these numbers is divisible by 6? (a) 4850 (b) 4856 (c) 4854 (d) 4865
13. 13.Which of these numbers is divisible by 11? (a) 1332 (b) 1320 (c) 1330 (d) 1316
14. 14.Which of these numbers is divisible by 6? (a) 4547 (b) 4548 (c) 4555 (d) 4550
15. 15.What is the remainder when 581 is divided by 8? (a) 6 (b) 4 (c) 5 (d) 3
16. 16.What is the units digit of 3^22? (a) 7 (b) 3 (c) 9 (d) 1
17. 17.What is the remainder when 535 is divided by 10? (a) 5 (b) 6 (c) 7 (d) 4
18. 18.What is the units digit of 8^52? (a) 4 (b) 2 (c) 8 (d) 6
19. 19.What is the units digit of 8^46? (a) 6 (b) 4 (c) 2 (d) 8
20. 20.What is the units digit of 3^45? (a) 3 (b) 1 (c) 7 (d) 9
21. 21.What is the remainder when 289 is divided by 10? (a) 8 (b) 9 (c) 7 (d) 1
22. 22.What is the smallest number you must add to 134 to make it divisible by 3? (a) 1 (b) 2
23. 23.What is the units digit of 7^35? (a) 9 (b) 3 (c) 1 (d) 7
24. 24.What is the remainder when 113 is divided by 3? (a) 1 (b) 0 (c) 2
25. 25.Which of these numbers is divisible by 3? (a) 1491 (b) 1493 (c) 1495 (d) 1489
26. 26.What is the units digit of 7^40? (a) 3 (b) 9 (c) 1 (d) 7
27. 27.What is the remainder when 156 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
28. 28.What is the units digit of 2^26? (a) 4 (b) 6 (c) 8 (d) 2
29. 29.What is the smallest number you must add to 510 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 6
30. 30.Which of these numbers is divisible by 4? (a) 1131 (b) 1123 (c) 1124 (d) 1126
31. 31.Which of these numbers is divisible by 3? (a) 1063 (b) 1058 (c) 1061 (d) 1059
32. 32.Which of these numbers is divisible by 6? (a) 2732 (b) 2728 (c) 2737 (d) 2730
33. 33.What is the remainder when 201 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
34. 34.Which of these numbers is divisible by 8? (a) 728 (b) 729 (c) 727 (d) 742
35. 35.What is the smallest number you must add to 79 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 5
36. 36.Which of these numbers is divisible by 8? (a) 4136 (b) 4145 (c) 4141 (d) 4133
37. 37.What is the remainder when 480 is divided by 9? (a) 4 (b) 2 (c) 3 (d) 6
38. 38.What is the remainder when 188 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 1
39. 39.Which of these numbers is divisible by 3? (a) 595 (b) 598 (c) 597 (d) 602
40. 40.What is the remainder when 191 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 0

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