Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 180 · 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 3^39? (a) 3 (b) 9 (c) 1 (d) 7
2. 2.What is the smallest number you must add to 193 to make it divisible by 9? (a) 6 (b) 5 (c) 7 (d) 4
3. 3.What is the remainder when 117 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
4. 4.What is the units digit of 9^31? (a) 7 (b) 9 (c) 8 (d) 1
5. 5.What is the remainder when 272 is divided by 7? (a) 1 (b) 5 (c) 4 (d) 6
6. 6.What is the units digit of 2^32? (a) 8 (b) 4 (c) 2 (d) 6
7. 7.Which of these numbers is divisible by 9? (a) 503 (b) 484 (c) 486 (d) 494
8. 8.What is the units digit of 4^20? (a) 5 (b) 4 (c) 7 (d) 6
9. 9.Which of these numbers is divisible by 3? (a) 1969 (b) 1972 (c) 1975 (d) 1971
10. 10.Which of these numbers is divisible by 11? (a) 6127 (b) 6137 (c) 6147 (d) 6120
11. 11.What is the smallest number you must add to 127 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 5
12. 12.Which of these numbers is divisible by 8? (a) 2135 (b) 2121 (c) 2143 (d) 2128
13. 13.What is the units digit of 3^10? (a) 1 (b) 3 (c) 7 (d) 9
14. 14.Which of these numbers is divisible by 3? (a) 2402 (b) 2400 (c) 2399 (d) 2405
15. 15.What is the smallest number you must add to 53 to make it divisible by 3? (a) 1 (b) 2
16. 16.What is the units digit of 8^15? (a) 2 (b) 4 (c) 8 (d) 6
17. 17.Which of these numbers is divisible by 4? (a) 2749 (b) 2755 (c) 2745 (d) 2748
18. 18.What is the remainder when 308 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
19. 19.What is the remainder when 311 is divided by 6? (a) 3 (b) 1 (c) 5 (d) 4
20. 20.What is the units digit of 2^27? (a) 4 (b) 2 (c) 6 (d) 8
21. 21.Which of these numbers is divisible by 4? (a) 3394 (b) 3398 (c) 3389 (d) 3392
22. 22.What is the units digit of 7^16? (a) 3 (b) 9 (c) 1 (d) 7
23. 23.Which of these numbers is divisible by 8? (a) 3039 (b) 3051 (c) 3040 (d) 3044
24. 24.What is the remainder when 555 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 2
25. 25.What is the remainder when 193 is divided by 11? (a) 5 (b) 8 (c) 6 (d) 7
26. 26.Which of these numbers is divisible by 6? (a) 1746 (b) 1748 (c) 1753 (d) 1741
27. 27.What is the remainder when 676 is divided by 10? (a) 6 (b) 7 (c) 5 (d) 4
28. 28.What is the units digit of 8^36? (a) 4 (b) 2 (c) 8 (d) 6
29. 29.What is the units digit of 9^51? (a) 9 (b) 7 (c) 8 (d) 1
30. 30.What is the smallest number you must add to 398 to make it divisible by 12? (a) 2 (b) 10 (c) 11 (d) 9
31. 31.What is the remainder when 392 is divided by 10? (a) 2 (b) 8 (c) 1 (d) 3
32. 32.Which of these numbers is divisible by 11? (a) 7392 (b) 7385 (c) 7406 (d) 7399
33. 33.Which of these numbers is divisible by 4? (a) 679 (b) 675 (c) 682 (d) 676
34. 34.What is the smallest number you must add to 484 to make it divisible by 8? (a) 6 (b) 4 (c) 5 (d) 3
35. 35.Which of these numbers is divisible by 3? (a) 514 (b) 518 (c) 521 (d) 516
36. 36.Which of these numbers is divisible by 8? (a) 5475 (b) 5482 (c) 5493 (d) 5480
37. 37.What is the units digit of 4^21? (a) 4 (b) 3 (c) 5 (d) 6
38. 38.What is the smallest number you must add to 249 to make it divisible by 4? (a) 1 (b) 2 (c) 3
39. 39.What is the remainder when 298 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
40. 40.What is the remainder when 382 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4

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