Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 794 · 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 183 to make it divisible by 4? (a) 3 (b) 2 (c) 1
2. 2.Which of these numbers is divisible by 11? (a) 4114 (b) 4126 (c) 4115 (d) 4112
3. 3.Which of these numbers is divisible by 8? (a) 2843 (b) 2832 (c) 2830 (d) 2838
4. 4.What is the units digit of 2^59? (a) 2 (b) 8 (c) 4 (d) 6
5. 5.Which of these numbers is divisible by 11? (a) 6889 (b) 6879 (c) 6875 (d) 6867
6. 6.What is the remainder when 233 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
7. 7.What is the units digit of 4^45? (a) 4 (b) 5 (c) 6 (d) 3
8. 8.Which of these numbers is divisible by 3? (a) 2308 (b) 2306 (c) 2302 (d) 2304
9. 9.What is the remainder when 766 is divided by 10? (a) 5 (b) 4 (c) 7 (d) 6
10. 10.What is the units digit of 3^39? (a) 3 (b) 7 (c) 1 (d) 9
11. 11.Which of these numbers is divisible by 8? (a) 3255 (b) 3247 (c) 3263 (d) 3248
12. 12.Which of these numbers is divisible by 11? (a) 2728 (b) 2734 (c) 2743 (d) 2719
13. 13.What is the remainder when 590 is divided by 8? (a) 5 (b) 6 (c) 7 (d) 2
14. 14.What is the remainder when 304 is divided by 6? (a) 5 (b) 2 (c) 3 (d) 4
15. 15.Which of these numbers is divisible by 9? (a) 5002 (b) 4983 (c) 4988 (d) 4986
16. 16.What is the units digit of 7^27? (a) 1 (b) 3 (c) 9 (d) 7
17. 17.What is the units digit of 7^56? (a) 7 (b) 9 (c) 3 (d) 1
18. 18.What is the smallest number you must add to 871 to make it divisible by 11? (a) 10 (b) 8 (c) 2 (d) 9
19. 19.What is the smallest number you must add to 68 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
20. 20.What is the smallest number you must add to 189 to make it divisible by 4? (a) 2 (b) 1 (c) 3
21. 21.What is the remainder when 407 is divided by 12? (a) 1 (b) 10 (c) 9 (d) 11
22. 22.What is the units digit of 8^46? (a) 2 (b) 8 (c) 6 (d) 4
23. 23.What is the remainder when 87 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
24. 24.Which of these numbers is divisible by 6? (a) 2828 (b) 2825 (c) 2820 (d) 2815
25. 25.Which of these numbers is divisible by 11? (a) 6148 (b) 6138 (c) 6158 (d) 6129
26. 26.What is the units digit of 7^44? (a) 9 (b) 7 (c) 1 (d) 3
27. 27.What is the units digit of 8^43? (a) 6 (b) 2 (c) 8 (d) 4
28. 28.Which of these numbers is divisible by 11? (a) 6182 (b) 6199 (c) 6185 (d) 6172
29. 29.What is the smallest number you must add to 437 to make it divisible by 9? (a) 4 (b) 6 (c) 3 (d) 5
30. 30.What is the smallest number you must add to 132 to make it divisible by 9? (a) 2 (b) 6 (c) 3 (d) 4
31. 31.What is the remainder when 74 is divided by 3? (a) 2 (b) 0 (c) 1
32. 32.Which of these numbers is divisible by 8? (a) 3460 (b) 3455 (c) 3456 (d) 3471
33. 33.Which of these numbers is divisible by 9? (a) 2082 (b) 2098 (c) 2091 (d) 2088
34. 34.What is the remainder when 257 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
35. 35.What is the remainder when 298 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
36. 36.Which of these numbers is divisible by 11? (a) 3240 (b) 3226 (c) 3234 (d) 3246
37. 37.What is the smallest number you must add to 359 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
38. 38.What is the remainder when 501 is divided by 9? (a) 6 (b) 5 (c) 7 (d) 3
39. 39.What is the remainder when 225 is divided by 12? (a) 3 (b) 9 (c) 8 (d) 10
40. 40.What is the units digit of 4^35? (a) 4 (b) 5 (c) 6 (d) 3

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