Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 263 · 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 210 to make it divisible by 11? (a) 1 (b) 9 (c) 10 (d) 8
2. 2.What is the remainder when 467 is divided by 10? (a) 3 (b) 8 (c) 7 (d) 6
3. 3.What is the remainder when 243 is divided by 8? (a) 2 (b) 5 (c) 3 (d) 4
4. 4.What is the smallest number you must add to 97 to make it divisible by 4? (a) 1 (b) 2 (c) 3
5. 5.What is the units digit of 7^34? (a) 1 (b) 3 (c) 7 (d) 9
6. 6.What is the units digit of 9^56? (a) 1 (b) 0 (c) 2 (d) 9
7. 7.Which of these numbers is divisible by 11? (a) 1258 (b) 1248 (c) 1254 (d) 1269
8. 8.What is the units digit of 8^45? (a) 4 (b) 8 (c) 6 (d) 2
9. 9.What is the remainder when 383 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 5
10. 10.What is the smallest number you must add to 346 to make it divisible by 8? (a) 6 (b) 5 (c) 2 (d) 7
11. 11.Which of these numbers is divisible by 3? (a) 1595 (b) 1596 (c) 1598 (d) 1601
12. 12.Which of these numbers is divisible by 4? (a) 2737 (b) 2741 (c) 2735 (d) 2736
13. 13.What is the remainder when 121 is divided by 8? (a) 0 (b) 7 (c) 2 (d) 1
14. 14.What is the smallest number you must add to 386 to make it divisible by 11? (a) 10 (b) 9 (c) 1 (d) 8
15. 15.What is the units digit of 2^52? (a) 4 (b) 6 (c) 8 (d) 2
16. 16.Which of these numbers is divisible by 6? (a) 1172 (b) 1176 (c) 1183 (d) 1179
17. 17.What is the units digit of 9^24? (a) 2 (b) 0 (c) 9 (d) 1
18. 18.What is the smallest number you must add to 324 to make it divisible by 11? (a) 5 (b) 8 (c) 6 (d) 7
19. 19.Which of these numbers is divisible by 9? (a) 7290 (b) 7291 (c) 7289 (d) 7304
20. 20.Which of these numbers is divisible by 8? (a) 6592 (b) 6601 (c) 6585 (d) 6599
21. 21.What is the units digit of 2^23? (a) 4 (b) 8 (c) 6 (d) 2
22. 22.Which of these numbers is divisible by 9? (a) 821 (b) 829 (c) 816 (d) 819
23. 23.Which of these numbers is divisible by 6? (a) 1592 (b) 1589 (c) 1583 (d) 1584
24. 24.What is the units digit of 4^23? (a) 6 (b) 5 (c) 3 (d) 4
25. 25.What is the units digit of 2^59? (a) 2 (b) 8 (c) 6 (d) 4
26. 26.What is the smallest number you must add to 303 to make it divisible by 7? (a) 5 (b) 4 (c) 2 (d) 6
27. 27.What is the units digit of 2^57? (a) 2 (b) 8 (c) 4 (d) 6
28. 28.What is the remainder when 146 is divided by 3? (a) 0 (b) 1 (c) 2
29. 29.What is the units digit of 8^34? (a) 8 (b) 2 (c) 4 (d) 6
30. 30.What is the smallest number you must add to 479 to make it divisible by 9? (a) 7 (b) 6 (c) 2 (d) 8
31. 31.What is the units digit of 3^48? (a) 9 (b) 7 (c) 1 (d) 3
32. 32.Which of these numbers is divisible by 8? (a) 4842 (b) 4832 (c) 4835 (d) 4827
33. 33.What is the units digit of 9^60? (a) 2 (b) 9 (c) 0 (d) 1
34. 34.What is the remainder when 305 is divided by 7? (a) 3 (b) 5 (c) 6 (d) 4
35. 35.What is the smallest number you must add to 75 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
36. 36.What is the units digit of 9^28? (a) 1 (b) 2 (c) 9 (d) 0
37. 37.Which of these numbers is divisible by 6? (a) 2095 (b) 2094 (c) 2105 (d) 2089
38. 38.Which of these numbers is divisible by 6? (a) 523 (b) 528 (c) 539 (d) 532
39. 39.Which of these numbers is divisible by 6? (a) 286 (b) 280 (c) 276 (d) 275
40. 40.What is the units digit of 3^19? (a) 3 (b) 7 (c) 9 (d) 1

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