Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 944 · 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 126 to make it divisible by 11? (a) 7 (b) 5 (c) 6 (d) 8
2. 2.What is the units digit of 9^34? (a) 0 (b) 9 (c) 2 (d) 1
3. 3.What is the remainder when 261 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
4. 4.Which of these numbers is divisible by 8? (a) 1958 (b) 1949 (c) 1966 (d) 1952
5. 5.Which of these numbers is divisible by 8? (a) 6549 (b) 6565 (c) 6553 (d) 6552
6. 6.What is the units digit of 4^14? (a) 6 (b) 5 (c) 7 (d) 4
7. 7.What is the units digit of 9^17? (a) 1 (b) 9 (c) 8 (d) 7
8. 8.What is the remainder when 149 is divided by 10? (a) 7 (b) 8 (c) 1 (d) 9
9. 9.Which of these numbers is divisible by 3? (a) 2491 (b) 2495 (c) 2493 (d) 2498
10. 10.What is the remainder when 422 is divided by 6? (a) 3 (b) 1 (c) 4 (d) 2
11. 11.Which of these numbers is divisible by 6? (a) 934 (b) 938 (c) 936 (d) 946
12. 12.Which of these numbers is divisible by 9? (a) 3292 (b) 3298 (c) 3307 (d) 3294
13. 13.What is the units digit of 7^24? (a) 7 (b) 1 (c) 3 (d) 9
14. 14.What is the smallest number you must add to 207 to make it divisible by 4? (a) 1 (b) 2 (c) 3
15. 15.What is the units digit of 8^19? (a) 8 (b) 6 (c) 4 (d) 2
16. 16.What is the remainder when 203 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
17. 17.What is the smallest number you must add to 786 to make it divisible by 10? (a) 4 (b) 5 (c) 3 (d) 6
18. 18.What is the smallest number you must add to 121 to make it divisible by 10? (a) 9 (b) 7 (c) 8 (d) 1
19. 19.What is the smallest number you must add to 458 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 3
20. 20.What is the units digit of 2^51? (a) 2 (b) 8 (c) 6 (d) 4
21. 21.Which of these numbers is divisible by 6? (a) 1879 (b) 1874 (c) 1888 (d) 1878
22. 22.What is the remainder when 258 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
23. 23.Which of these numbers is divisible by 9? (a) 3276 (b) 3283 (c) 3275 (d) 3289
24. 24.What is the remainder when 591 is divided by 11? (a) 9 (b) 8 (c) 7 (d) 3
25. 25.What is the units digit of 8^31? (a) 2 (b) 4 (c) 6 (d) 8
26. 26.Which of these numbers is divisible by 11? (a) 6258 (b) 6248 (c) 6246 (d) 6265
27. 27.Which of these numbers is divisible by 3? (a) 932 (b) 933 (c) 935 (d) 937
28. 28.What is the smallest number you must add to 70 to make it divisible by 3? (a) 1 (b) 2
29. 29.Which of these numbers is divisible by 8? (a) 5056 (b) 5053 (c) 5068 (d) 5061
30. 30.What is the units digit of 8^23? (a) 8 (b) 4 (c) 6 (d) 2
31. 31.What is the smallest number you must add to 49 to make it divisible by 4? (a) 2 (b) 3 (c) 1
32. 32.What is the smallest number you must add to 199 to make it divisible by 12? (a) 4 (b) 6 (c) 7 (d) 5
33. 33.What is the units digit of 2^34? (a) 6 (b) 8 (c) 2 (d) 4
34. 34.What is the smallest number you must add to 471 to make it divisible by 6? (a) 5 (b) 4 (c) 2 (d) 3
35. 35.What is the units digit of 9^23? (a) 8 (b) 1 (c) 9 (d) 7
36. 36.Which of these numbers is divisible by 8? (a) 5238 (b) 5223 (c) 5224 (d) 5228
37. 37.What is the units digit of 4^42? (a) 7 (b) 5 (c) 6 (d) 4
38. 38.What is the smallest number you must add to 95 to make it divisible by 7? (a) 5 (b) 2 (c) 4 (d) 3
39. 39.Which of these numbers is divisible by 3? (a) 1367 (b) 1364 (c) 1365 (d) 1369
40. 40.What is the smallest number you must add to 305 to make it divisible by 4? (a) 2 (b) 1 (c) 3

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