Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 985 · 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 178 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
2. 2.What is the units digit of 4^18? (a) 5 (b) 7 (c) 4 (d) 6
3. 3.What is the remainder when 427 is divided by 10? (a) 6 (b) 3 (c) 7 (d) 8
4. 4.Which of these numbers is divisible by 8? (a) 4693 (b) 4680 (c) 4679 (d) 4684
5. 5.What is the remainder when 619 is divided by 8? (a) 2 (b) 3 (c) 5 (d) 4
6. 6.What is the remainder when 122 is divided by 9? (a) 7 (b) 4 (c) 6 (d) 5
7. 7.What is the remainder when 279 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
8. 8.What is the remainder when 107 is divided by 8? (a) 2 (b) 4 (c) 3 (d) 5
9. 9.What is the units digit of 4^15? (a) 3 (b) 6 (c) 4 (d) 5
10. 10.Which of these numbers is divisible by 6? (a) 2063 (b) 2064 (c) 2065 (d) 2073
11. 11.Which of these numbers is divisible by 6? (a) 5035 (b) 5042 (c) 5048 (d) 5040
12. 12.Which of these numbers is divisible by 4? (a) 905 (b) 911 (c) 904 (d) 902
13. 13.What is the units digit of 4^31? (a) 5 (b) 3 (c) 4 (d) 6
14. 14.What is the smallest number you must add to 646 to make it divisible by 12? (a) 10 (b) 3 (c) 2 (d) 1
15. 15.What is the units digit of 7^38? (a) 3 (b) 1 (c) 9 (d) 7
16. 16.Which of these numbers is divisible by 11? (a) 5648 (b) 5633 (c) 5643 (d) 5655
17. 17.What is the remainder when 303 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
18. 18.Which of these numbers is divisible by 11? (a) 4545 (b) 4532 (c) 4530 (d) 4538
19. 19.What is the remainder when 340 is divided by 9? (a) 2 (b) 6 (c) 8 (d) 7
20. 20.What is the smallest number you must add to 101 to make it divisible by 4? (a) 2 (b) 3 (c) 1
21. 21.Which of these numbers is divisible by 4? (a) 634 (b) 631 (c) 632 (d) 639
22. 22.What is the remainder when 79 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
23. 23.What is the units digit of 8^41? (a) 6 (b) 4 (c) 2 (d) 8
24. 24.What is the units digit of 2^24? (a) 8 (b) 6 (c) 4 (d) 2
25. 25.Which of these numbers is divisible by 9? (a) 4485 (b) 4501 (c) 4497 (d) 4491
26. 26.What is the remainder when 172 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
27. 27.Which of these numbers is divisible by 6? (a) 4412 (b) 4401 (c) 4404 (d) 4405
28. 28.Which of these numbers is divisible by 3? (a) 1920 (b) 1918 (c) 1925 (d) 1921
29. 29.What is the units digit of 2^19? (a) 8 (b) 2 (c) 6 (d) 4
30. 30.Which of these numbers is divisible by 8? (a) 4998 (b) 5012 (c) 5000 (d) 5004
31. 31.What is the smallest number you must add to 261 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
32. 32.What is the remainder when 507 is divided by 7? (a) 2 (b) 3 (c) 4 (d) 5
33. 33.What is the smallest number you must add to 232 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
34. 34.What is the units digit of 3^22? (a) 1 (b) 3 (c) 7 (d) 9
35. 35.Which of these numbers is divisible by 4? (a) 2946 (b) 2940 (c) 2939 (d) 2943
36. 36.What is the units digit of 4^14? (a) 7 (b) 4 (c) 6 (d) 5
37. 37.Which of these numbers is divisible by 9? (a) 5161 (b) 5166 (c) 5168 (d) 5183
38. 38.What is the remainder when 459 is divided by 6? (a) 5 (b) 3 (c) 2 (d) 4
39. 39.What is the units digit of 2^53? (a) 2 (b) 6 (c) 4 (d) 8
40. 40.What is the smallest number you must add to 193 to make it divisible by 3? (a) 1 (b) 2

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