Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 8 · 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 remainder when 653 is divided by 10? (a) 3 (b) 7 (c) 4 (d) 2
2. 2.Which of these numbers is divisible by 9? (a) 1738 (b) 1737 (c) 1749 (d) 1730
3. 3.Which of these numbers is divisible by 8? (a) 2298 (b) 2295 (c) 2296 (d) 2311
4. 4.Which of these numbers is divisible by 9? (a) 7114 (b) 7130 (c) 7127 (d) 7119
5. 5.What is the units digit of 8^19? (a) 6 (b) 2 (c) 8 (d) 4
6. 6.Which of these numbers is divisible by 6? (a) 1926 (b) 1921 (c) 1937 (d) 1929
7. 7.What is the units digit of 9^18? (a) 2 (b) 9 (c) 1 (d) 0
8. 8.What is the units digit of 7^13? (a) 1 (b) 7 (c) 9 (d) 3
9. 9.Which of these numbers is divisible by 9? (a) 1977 (b) 1969 (c) 1984 (d) 1971
10. 10.What is the remainder when 205 is divided by 3? (a) 2 (b) 0 (c) 1
11. 11.Which of these numbers is divisible by 4? (a) 3160 (b) 3165 (c) 3157 (d) 3163
12. 12.What is the remainder when 862 is divided by 11? (a) 5 (b) 7 (c) 4 (d) 3
13. 13.What is the smallest number you must add to 233 to make it divisible by 3? (a) 2 (b) 1
14. 14.What is the remainder when 450 is divided by 11? (a) 10 (b) 1 (c) 9 (d) 8
15. 15.What is the units digit of 2^16? (a) 6 (b) 2 (c) 8 (d) 4
16. 16.Which of these numbers is divisible by 9? (a) 4419 (b) 4422 (c) 4418 (d) 4432
17. 17.What is the units digit of 7^46? (a) 3 (b) 1 (c) 7 (d) 9
18. 18.Which of these numbers is divisible by 3? (a) 2158 (b) 2156 (c) 2157 (d) 2161
19. 19.What is the remainder when 211 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
20. 20.What is the remainder when 154 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
21. 21.Which of these numbers is divisible by 4? (a) 2319 (b) 2315 (c) 2316 (d) 2321
22. 22.What is the units digit of 4^45? (a) 4 (b) 3 (c) 6 (d) 5
23. 23.Which of these numbers is divisible by 4? (a) 2773 (b) 2770 (c) 2767 (d) 2768
24. 24.What is the units digit of 2^43? (a) 6 (b) 2 (c) 8 (d) 4
25. 25.What is the remainder when 107 is divided by 10? (a) 6 (b) 7 (c) 8 (d) 3
26. 26.What is the units digit of 2^14? (a) 6 (b) 4 (c) 2 (d) 8
27. 27.Which of these numbers is divisible by 3? (a) 568 (b) 571 (c) 566 (d) 567
28. 28.What is the remainder when 31 is divided by 3? (a) 1 (b) 2 (c) 0
29. 29.What is the smallest number you must add to 296 to make it divisible by 7? (a) 6 (b) 2 (c) 4 (d) 5
30. 30.What is the units digit of 3^34? (a) 7 (b) 9 (c) 3 (d) 1
31. 31.What is the units digit of 3^53? (a) 9 (b) 7 (c) 1 (d) 3
32. 32.What is the smallest number you must add to 589 to make it divisible by 11? (a) 6 (b) 4 (c) 5 (d) 7
33. 33.Which of these numbers is divisible by 6? (a) 3955 (b) 3947 (c) 3948 (d) 3949
34. 34.Which of these numbers is divisible by 4? (a) 1249 (b) 1254 (c) 1248 (d) 1246
35. 35.What is the units digit of 3^19? (a) 1 (b) 9 (c) 3 (d) 7
36. 36.Which of these numbers is divisible by 3? (a) 2113 (b) 2110 (c) 2109 (d) 2108
37. 37.What is the units digit of 2^42? (a) 6 (b) 2 (c) 8 (d) 4
38. 38.What is the smallest number you must add to 363 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
39. 39.What is the smallest number you must add to 451 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
40. 40.What is the remainder when 395 is divided by 8? (a) 5 (b) 2 (c) 3 (d) 4

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