Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 823 · 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 4^16? (a) 5 (b) 7 (c) 6 (d) 4
2. 2.Which of these numbers is divisible by 8? (a) 3993 (b) 3989 (c) 4005 (d) 3992
3. 3.What is the smallest number you must add to 50 to make it divisible by 3? (a) 2 (b) 1
4. 4.What is the units digit of 3^58? (a) 9 (b) 1 (c) 3 (d) 7
5. 5.Which of these numbers is divisible by 8? (a) 832 (b) 837 (c) 826 (d) 842
6. 6.Which of these numbers is divisible by 6? (a) 1656 (b) 1655 (c) 1660 (d) 1666
7. 7.What is the remainder when 284 is divided by 11? (a) 8 (b) 2 (c) 9 (d) 10
8. 8.Which of these numbers is divisible by 8? (a) 1648 (b) 1642 (c) 1659 (d) 1649
9. 9.Which of these numbers is divisible by 4? (a) 1109 (b) 1106 (c) 1108 (d) 1113
10. 10.What is the units digit of 7^23? (a) 7 (b) 9 (c) 1 (d) 3
11. 11.Which of these numbers is divisible by 8? (a) 691 (b) 677 (c) 680 (d) 685
12. 12.Which of these numbers is divisible by 11? (a) 9113 (b) 9120 (c) 9102 (d) 9108
13. 13.What is the units digit of 2^37? (a) 8 (b) 2 (c) 4 (d) 6
14. 14.What is the remainder when 523 is divided by 12? (a) 6 (b) 8 (c) 7 (d) 5
15. 15.Which of these numbers is divisible by 9? (a) 1986 (b) 1979 (c) 1971 (d) 1967
16. 16.What is the remainder when 204 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
17. 17.What is the remainder when 392 is divided by 11? (a) 8 (b) 6 (c) 4 (d) 7
18. 18.Which of these numbers is divisible by 8? (a) 6306 (b) 6304 (c) 6300 (d) 6314
19. 19.What is the units digit of 2^34? (a) 6 (b) 4 (c) 2 (d) 8
20. 20.What is the remainder when 343 is divided by 11? (a) 3 (b) 1 (c) 9 (d) 2
21. 21.Which of these numbers is divisible by 3? (a) 1124 (b) 1127 (c) 1122 (d) 1121
22. 22.Which of these numbers is divisible by 4? (a) 1450 (b) 1443 (c) 1444 (d) 1446
23. 23.Which of these numbers is divisible by 6? (a) 1784 (b) 1774 (c) 1776 (d) 1777
24. 24.Which of these numbers is divisible by 3? (a) 766 (b) 762 (c) 761 (d) 764
25. 25.Which of these numbers is divisible by 4? (a) 859 (b) 865 (c) 860 (d) 861
26. 26.Which of these numbers is divisible by 8? (a) 1520 (b) 1529 (c) 1522 (d) 1519
27. 27.What is the units digit of 2^48? (a) 2 (b) 4 (c) 6 (d) 8
28. 28.What is the smallest number you must add to 69 to make it divisible by 4? (a) 1 (b) 2 (c) 3
29. 29.What is the smallest number you must add to 213 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
30. 30.What is the remainder when 282 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
31. 31.Which of these numbers is divisible by 9? (a) 624 (b) 612 (c) 613 (d) 610
32. 32.What is the units digit of 2^53? (a) 6 (b) 2 (c) 8 (d) 4
33. 33.What is the smallest number you must add to 91 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 1
34. 34.What is the smallest number you must add to 97 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 1
35. 35.What is the remainder when 209 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
36. 36.What is the smallest number you must add to 179 to make it divisible by 3? (a) 1 (b) 2
37. 37.What is the smallest number you must add to 236 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
38. 38.Which of these numbers is divisible by 3? (a) 1243 (b) 1241 (c) 1247 (d) 1242
39. 39.What is the smallest number you must add to 136 to make it divisible by 3? (a) 1 (b) 2
40. 40.Which of these numbers is divisible by 9? (a) 3034 (b) 3032 (c) 3044 (d) 3033

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