Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 137 · 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 263 to make it divisible by 4? (a) 2 (b) 3 (c) 1
2. 2.What is the smallest number you must add to 143 to make it divisible by 7? (a) 6 (b) 5 (c) 3 (d) 4
3. 3.What is the remainder when 464 is divided by 6? (a) 2 (b) 3 (c) 1 (d) 4
4. 4.What is the remainder when 287 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
5. 5.What is the units digit of 3^39? (a) 3 (b) 7 (c) 1 (d) 9
6. 6.What is the units digit of 4^16? (a) 4 (b) 7 (c) 5 (d) 6
7. 7.What is the remainder when 410 is divided by 8? (a) 6 (b) 2 (c) 1 (d) 3
8. 8.What is the remainder when 436 is divided by 11? (a) 8 (b) 7 (c) 6 (d) 4
9. 9.What is the remainder when 213 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
10. 10.What is the units digit of 4^49? (a) 6 (b) 5 (c) 4 (d) 3
11. 11.What is the smallest number you must add to 121 to make it divisible by 4? (a) 2 (b) 1 (c) 3
12. 12.Which of these numbers is divisible by 4? (a) 2982 (b) 2977 (c) 2975 (d) 2976
13. 13.What is the smallest number you must add to 400 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 1
14. 14.What is the remainder when 955 is divided by 12? (a) 7 (b) 8 (c) 6 (d) 5
15. 15.Which of these numbers is divisible by 11? (a) 4070 (b) 4066 (c) 4072 (d) 4085
16. 16.What is the remainder when 272 is divided by 9? (a) 1 (b) 2 (c) 7 (d) 3
17. 17.Which of these numbers is divisible by 3? (a) 654 (b) 653 (c) 656 (d) 659
18. 18.What is the units digit of 3^33? (a) 3 (b) 7 (c) 1 (d) 9
19. 19.What is the remainder when 130 is divided by 9? (a) 3 (b) 4 (c) 6 (d) 5
20. 20.What is the remainder when 394 is divided by 9? (a) 2 (b) 8 (c) 6 (d) 7
21. 21.What is the units digit of 8^20? (a) 2 (b) 8 (c) 4 (d) 6
22. 22.Which of these numbers is divisible by 3? (a) 603 (b) 602 (c) 607 (d) 604
23. 23.Which of these numbers is divisible by 4? (a) 1621 (b) 1625 (c) 1630 (d) 1624
24. 24.What is the remainder when 193 is divided by 3? (a) 1 (b) 2 (c) 0
25. 25.What is the smallest number you must add to 184 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
26. 26.What is the remainder when 681 is divided by 11? (a) 10 (b) 1 (c) 8 (d) 9
27. 27.Which of these numbers is divisible by 8? (a) 7060 (b) 7048 (c) 7051 (d) 7041
28. 28.What is the remainder when 61 is divided by 3? (a) 1 (b) 0 (c) 2
29. 29.What is the units digit of 7^18? (a) 7 (b) 3 (c) 1 (d) 9
30. 30.Which of these numbers is divisible by 3? (a) 2613 (b) 2614 (c) 2612 (d) 2617
31. 31.What is the smallest number you must add to 673 to make it divisible by 10? (a) 3 (b) 8 (c) 6 (d) 7
32. 32.What is the units digit of 9^26? (a) 0 (b) 2 (c) 1 (d) 9
33. 33.What is the units digit of 7^28? (a) 7 (b) 9 (c) 1 (d) 3
34. 34.What is the smallest number you must add to 535 to make it divisible by 8? (a) 3 (b) 7 (c) 2 (d) 1
35. 35.What is the smallest number you must add to 293 to make it divisible by 4? (a) 2 (b) 3 (c) 1
36. 36.What is the units digit of 2^46? (a) 6 (b) 8 (c) 4 (d) 2
37. 37.What is the units digit of 2^21? (a) 4 (b) 6 (c) 2 (d) 8
38. 38.What is the remainder when 693 is divided by 10? (a) 3 (b) 7 (c) 4 (d) 2
39. 39.What is the smallest number you must add to 439 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
40. 40.What is the remainder when 253 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3

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