Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 601 · 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 633 is divided by 9? (a) 2 (b) 4 (c) 6 (d) 3
2. 2.What is the units digit of 4^35? (a) 4 (b) 5 (c) 3 (d) 6
3. 3.What is the remainder when 232 is divided by 12? (a) 8 (b) 4 (c) 5 (d) 3
4. 4.What is the smallest number you must add to 71 to make it divisible by 4? (a) 2 (b) 3 (c) 1
5. 5.What is the remainder when 461 is divided by 10? (a) 1 (b) 0 (c) 2 (d) 9
6. 6.Which of these numbers is divisible by 8? (a) 498 (b) 496 (c) 491 (d) 506
7. 7.What is the smallest number you must add to 215 to make it divisible by 7? (a) 5 (b) 3 (c) 2 (d) 1
8. 8.What is the units digit of 2^54? (a) 8 (b) 4 (c) 6 (d) 2
9. 9.Which of these numbers is divisible by 4? (a) 2776 (b) 2777 (c) 2782 (d) 2775
10. 10.What is the smallest number you must add to 164 to make it divisible by 3? (a) 2 (b) 1
11. 11.Which of these numbers is divisible by 3? (a) 1622 (b) 1623 (c) 1624 (d) 1627
12. 12.What is the units digit of 9^50? (a) 1 (b) 2 (c) 9 (d) 0
13. 13.What is the remainder when 61 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
14. 14.Which of these numbers is divisible by 3? (a) 224 (b) 225 (c) 230 (d) 227
15. 15.What is the units digit of 4^40? (a) 6 (b) 4 (c) 7 (d) 5
16. 16.What is the remainder when 264 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
17. 17.What is the units digit of 4^48? (a) 4 (b) 5 (c) 6 (d) 7
18. 18.What is the units digit of 2^11? (a) 2 (b) 6 (c) 8 (d) 4
19. 19.What is the units digit of 4^53? (a) 6 (b) 4 (c) 3 (d) 5
20. 20.What is the smallest number you must add to 740 to make it divisible by 11? (a) 9 (b) 7 (c) 3 (d) 8
21. 21.What is the remainder when 129 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
22. 22.What is the remainder when 444 is divided by 10? (a) 6 (b) 3 (c) 5 (d) 4
23. 23.What is the units digit of 3^60? (a) 1 (b) 3 (c) 7 (d) 9
24. 24.What is the units digit of 2^41? (a) 8 (b) 6 (c) 4 (d) 2
25. 25.What is the smallest number you must add to 324 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
26. 26.What is the smallest number you must add to 163 to make it divisible by 3? (a) 2 (b) 1
27. 27.Which of these numbers is divisible by 9? (a) 473 (b) 461 (c) 459 (d) 452
28. 28.What is the smallest number you must add to 57 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
29. 29.What is the remainder when 408 is divided by 9? (a) 6 (b) 3 (c) 4 (d) 2
30. 30.What is the units digit of 7^46? (a) 7 (b) 9 (c) 1 (d) 3
31. 31.What is the units digit of 8^17? (a) 6 (b) 4 (c) 2 (d) 8
32. 32.What is the smallest number you must add to 129 to make it divisible by 8? (a) 7 (b) 6 (c) 1 (d) 5
33. 33.Which of these numbers is divisible by 8? (a) 2978 (b) 2989 (c) 2984 (d) 2996
34. 34.What is the remainder when 581 is divided by 11? (a) 8 (b) 10 (c) 2 (d) 9
35. 35.Which of these numbers is divisible by 6? (a) 4950 (b) 4960 (c) 4945 (d) 4955
36. 36.What is the remainder when 130 is divided by 11? (a) 2 (b) 10 (c) 8 (d) 9
37. 37.Which of these numbers is divisible by 11? (a) 1980 (b) 1977 (c) 1998 (d) 1984
38. 38.Which of these numbers is divisible by 9? (a) 1840 (b) 1836 (c) 1834 (d) 1847
39. 39.What is the remainder when 646 is divided by 11? (a) 9 (b) 3 (c) 8 (d) 7
40. 40.Which of these numbers is divisible by 11? (a) 9623 (b) 9628 (c) 9614 (d) 9611

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