Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 74 · 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 2^21? (a) 2 (b) 6 (c) 8 (d) 4
2. 2.What is the units digit of 8^11? (a) 4 (b) 2 (c) 6 (d) 8
3. 3.Which of these numbers is divisible by 8? (a) 4770 (b) 4759 (c) 4761 (d) 4760
4. 4.Which of these numbers is divisible by 3? (a) 251 (b) 247 (c) 246 (d) 245
5. 5.What is the smallest number you must add to 186 to make it divisible by 4? (a) 2 (b) 3 (c) 1
6. 6.Which of these numbers is divisible by 11? (a) 5251 (b) 5232 (c) 5236 (d) 5245
7. 7.Which of these numbers is divisible by 11? (a) 4070 (b) 4090 (c) 4071 (d) 4068
8. 8.What is the units digit of 3^37? (a) 3 (b) 1 (c) 7 (d) 9
9. 9.Which of these numbers is divisible by 8? (a) 2408 (b) 2402 (c) 2409 (d) 2421
10. 10.What is the units digit of 7^10? (a) 3 (b) 1 (c) 9 (d) 7
11. 11.Which of these numbers is divisible by 6? (a) 1062 (b) 1060 (c) 1069 (d) 1066
12. 12.What is the remainder when 356 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 1
13. 13.What is the remainder when 246 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
14. 14.Which of these numbers is divisible by 4? (a) 591 (b) 592 (c) 594 (d) 598
15. 15.Which of these numbers is divisible by 6? (a) 1590 (b) 1600 (c) 1592 (d) 1588
16. 16.What is the smallest number you must add to 506 to make it divisible by 10? (a) 5 (b) 4 (c) 3 (d) 6
17. 17.What is the units digit of 3^21? (a) 3 (b) 7 (c) 1 (d) 9
18. 18.What is the smallest number you must add to 107 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 2
19. 19.Which of these numbers is divisible by 9? (a) 7579 (b) 7587 (c) 7595 (d) 7601
20. 20.What is the units digit of 2^48? (a) 8 (b) 2 (c) 6 (d) 4
21. 21.What is the smallest number you must add to 115 to make it divisible by 4? (a) 2 (b) 3 (c) 1
22. 22.What is the smallest number you must add to 136 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 1
23. 23.What is the units digit of 2^26? (a) 8 (b) 2 (c) 4 (d) 6
24. 24.What is the smallest number you must add to 585 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 7
25. 25.What is the units digit of 9^33? (a) 8 (b) 1 (c) 7 (d) 9
26. 26.What is the smallest number you must add to 179 to make it divisible by 4? (a) 3 (b) 2 (c) 1
27. 27.What is the units digit of 3^35? (a) 3 (b) 9 (c) 7 (d) 1
28. 28.What is the units digit of 3^45? (a) 3 (b) 9 (c) 7 (d) 1
29. 29.Which of these numbers is divisible by 9? (a) 548 (b) 555 (c) 560 (d) 549
30. 30.Which of these numbers is divisible by 4? (a) 3565 (b) 3563 (c) 3569 (d) 3564
31. 31.Which of these numbers is divisible by 4? (a) 1980 (b) 1979 (c) 1981 (d) 1986
32. 32.What is the smallest number you must add to 121 to make it divisible by 3? (a) 2 (b) 1
33. 33.What is the smallest number you must add to 335 to make it divisible by 7? (a) 6 (b) 3 (c) 1 (d) 2
34. 34.Which of these numbers is divisible by 6? (a) 2125 (b) 2123 (c) 2115 (d) 2118
35. 35.What is the remainder when 479 is divided by 8? (a) 6 (b) 1 (c) 5 (d) 7
36. 36.What is the smallest number you must add to 165 to make it divisible by 4? (a) 2 (b) 3 (c) 1
37. 37.What is the units digit of 4^24? (a) 5 (b) 7 (c) 4 (d) 6
38. 38.What is the smallest number you must add to 487 to make it divisible by 11? (a) 9 (b) 3 (c) 8 (d) 7
39. 39.Which of these numbers is divisible by 8? (a) 2086 (b) 2078 (c) 2080 (d) 2090
40. 40.What is the remainder when 788 is divided by 10? (a) 2 (b) 9 (c) 8 (d) 7

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