Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 118 · 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 220 to make it divisible by 8? (a) 5 (b) 4 (c) 6 (d) 3
2. 2.What is the remainder when 759 is divided by 10? (a) 9 (b) 1 (c) 8 (d) 7
3. 3.What is the remainder when 399 is divided by 8? (a) 7 (b) 1 (c) 5 (d) 6
4. 4.What is the smallest number you must add to 53 to make it divisible by 3? (a) 2 (b) 1
5. 5.Which of these numbers is divisible by 4? (a) 2416 (b) 2422 (c) 2415 (d) 2419
6. 6.What is the smallest number you must add to 242 to make it divisible by 3? (a) 1 (b) 2
7. 7.What is the smallest number you must add to 389 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 2
8. 8.Which of these numbers is divisible by 3? (a) 2515 (b) 2510 (c) 2511 (d) 2512
9. 9.What is the smallest number you must add to 546 to make it divisible by 10? (a) 3 (b) 6 (c) 4 (d) 5
10. 10.What is the smallest number you must add to 113 to make it divisible by 4? (a) 2 (b) 3 (c) 1
11. 11.What is the smallest number you must add to 515 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
12. 12.Which of these numbers is divisible by 3? (a) 2094 (b) 2098 (c) 2092 (d) 2096
13. 13.What is the units digit of 7^34? (a) 1 (b) 3 (c) 9 (d) 7
14. 14.What is the units digit of 2^39? (a) 6 (b) 4 (c) 8 (d) 2
15. 15.What is the units digit of 4^46? (a) 6 (b) 4 (c) 7 (d) 5
16. 16.Which of these numbers is divisible by 3? (a) 1093 (b) 1088 (c) 1089 (d) 1091
17. 17.What is the units digit of 4^17? (a) 3 (b) 6 (c) 5 (d) 4
18. 18.What is the units digit of 4^48? (a) 5 (b) 6 (c) 4 (d) 7
19. 19.What is the remainder when 500 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 3
20. 20.What is the remainder when 669 is divided by 12? (a) 8 (b) 10 (c) 9 (d) 3
21. 21.Which of these numbers is divisible by 6? (a) 4817 (b) 4819 (c) 4829 (d) 4818
22. 22.Which of these numbers is divisible by 9? (a) 7044 (b) 7038 (c) 7031 (d) 7049
23. 23.What is the smallest number you must add to 699 to make it divisible by 12? (a) 3 (b) 10 (c) 8 (d) 9
24. 24.What is the units digit of 3^33? (a) 7 (b) 1 (c) 3 (d) 9
25. 25.What is the units digit of 9^42? (a) 9 (b) 0 (c) 2 (d) 1
26. 26.What is the smallest number you must add to 77 to make it divisible by 3? (a) 2 (b) 1
27. 27.What is the smallest number you must add to 281 to make it divisible by 11? (a) 5 (b) 4 (c) 6 (d) 7
28. 28.Which of these numbers is divisible by 8? (a) 4537 (b) 4536 (c) 4547 (d) 4535
29. 29.Which of these numbers is divisible by 6? (a) 4320 (b) 4330 (c) 4323 (d) 4318
30. 30.Which of these numbers is divisible by 6? (a) 4359 (b) 4369 (c) 4362 (d) 4365
31. 31.What is the remainder when 454 is divided by 6? (a) 3 (b) 5 (c) 2 (d) 4
32. 32.What is the smallest number you must add to 153 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
33. 33.What is the smallest number you must add to 149 to make it divisible by 3? (a) 2 (b) 1
34. 34.What is the units digit of 7^35? (a) 7 (b) 1 (c) 3 (d) 9
35. 35.What is the remainder when 439 is divided by 9? (a) 2 (b) 7 (c) 6 (d) 8
36. 36.Which of these numbers is divisible by 3? (a) 1273 (b) 1271 (c) 1272 (d) 1277
37. 37.Which of these numbers is divisible by 8? (a) 2229 (b) 2244 (c) 2237 (d) 2232
38. 38.Which of these numbers is divisible by 3? (a) 2584 (b) 2581 (c) 2583 (d) 2588
39. 39.What is the smallest number you must add to 384 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
40. 40.What is the units digit of 3^13? (a) 7 (b) 3 (c) 9 (d) 1

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