Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 649 · 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 8^30? (a) 4 (b) 8 (c) 6 (d) 2
2. 2.What is the smallest number you must add to 172 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 2
3. 3.What is the remainder when 478 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
4. 4.What is the remainder when 110 is divided by 8? (a) 6 (b) 2 (c) 5 (d) 7
5. 5.What is the remainder when 274 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 5
6. 6.Which of these numbers is divisible by 4? (a) 1303 (b) 1298 (c) 1295 (d) 1296
7. 7.What is the units digit of 7^33? (a) 1 (b) 7 (c) 3 (d) 9
8. 8.Which of these numbers is divisible by 6? (a) 1729 (b) 1722 (c) 1721 (d) 1723
9. 9.What is the units digit of 9^13? (a) 1 (b) 7 (c) 9 (d) 8
10. 10.What is the remainder when 238 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
11. 11.Which of these numbers is divisible by 4? (a) 1907 (b) 1904 (c) 1902 (d) 1911
12. 12.Which of these numbers is divisible by 4? (a) 3095 (b) 3091 (c) 3086 (d) 3088
13. 13.Which of these numbers is divisible by 8? (a) 1248 (b) 1251 (c) 1257 (d) 1246
14. 14.Which of these numbers is divisible by 11? (a) 4134 (b) 4156 (c) 4141 (d) 4136
15. 15.Which of these numbers is divisible by 11? (a) 5657 (b) 5640 (c) 5643 (d) 5647
16. 16.What is the smallest number you must add to 316 to make it divisible by 7? (a) 6 (b) 4 (c) 1 (d) 5
17. 17.Which of these numbers is divisible by 11? (a) 520 (b) 515 (c) 536 (d) 517
18. 18.Which of these numbers is divisible by 3? (a) 1582 (b) 1584 (c) 1588 (d) 1585
19. 19.What is the smallest number you must add to 330 to make it divisible by 7? (a) 6 (b) 5 (c) 1 (d) 4
20. 20.Which of these numbers is divisible by 11? (a) 9434 (b) 9423 (c) 9411 (d) 9416
21. 21.Which of these numbers is divisible by 3? (a) 1948 (b) 1947 (c) 1946 (d) 1952
22. 22.What is the remainder when 663 is divided by 12? (a) 2 (b) 9 (c) 4 (d) 3
23. 23.What is the smallest number you must add to 173 to make it divisible by 4? (a) 1 (b) 2 (c) 3
24. 24.What is the smallest number you must add to 187 to make it divisible by 8? (a) 5 (b) 4 (c) 6 (d) 3
25. 25.What is the units digit of 8^25? (a) 4 (b) 2 (c) 8 (d) 6
26. 26.What is the remainder when 312 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
27. 27.Which of these numbers is divisible by 9? (a) 4525 (b) 4527 (c) 4533 (d) 4541
28. 28.What is the units digit of 2^17? (a) 6 (b) 4 (c) 8 (d) 2
29. 29.What is the smallest number you must add to 297 to make it divisible by 8? (a) 5 (b) 6 (c) 1 (d) 7
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 4^16? (a) 6 (b) 7 (c) 5 (d) 4
32. 32.Which of these numbers is divisible by 11? (a) 3869 (b) 3856 (c) 3861 (d) 3877
33. 33.Which of these numbers is divisible by 6? (a) 1503 (b) 1500 (c) 1509 (d) 1499
34. 34.What is the smallest number you must add to 385 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 1
35. 35.Which of these numbers is divisible by 3? (a) 2136 (b) 2140 (c) 2138 (d) 2134
36. 36.Which of these numbers is divisible by 6? (a) 2690 (b) 2697 (c) 2688 (d) 2685
37. 37.What is the units digit of 9^45? (a) 9 (b) 7 (c) 1 (d) 8
38. 38.Which of these numbers is divisible by 4? (a) 1653 (b) 1663 (c) 1657 (d) 1656
39. 39.What is the smallest number you must add to 533 to make it divisible by 7? (a) 4 (b) 6 (c) 1 (d) 5
40. 40.What is the remainder when 170 is divided by 12? (a) 2 (b) 1 (c) 3 (d) 10

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