Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 973 · 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 3^33? (a) 7 (b) 9 (c) 1 (d) 3
2. 2.Which of these numbers is divisible by 3? (a) 2497 (b) 2496 (c) 2495 (d) 2501
3. 3.What is the smallest number you must add to 230 to make it divisible by 4? (a) 1 (b) 2 (c) 3
4. 4.What is the remainder when 434 is divided by 11? (a) 6 (b) 4 (c) 7 (d) 5
5. 5.What is the units digit of 8^52? (a) 8 (b) 6 (c) 2 (d) 4
6. 6.What is the remainder when 176 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
7. 7.What is the remainder when 381 is divided by 11? (a) 4 (b) 8 (c) 7 (d) 6
8. 8.What is the units digit of 8^15? (a) 8 (b) 2 (c) 6 (d) 4
9. 9.What is the smallest number you must add to 358 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 6
10. 10.What is the units digit of 2^31? (a) 6 (b) 4 (c) 8 (d) 2
11. 11.What is the smallest number you must add to 56 to make it divisible by 3? (a) 2 (b) 1
12. 12.Which of these numbers is divisible by 8? (a) 5683 (b) 5680 (c) 5689 (d) 5674
13. 13.What is the remainder when 400 is divided by 11? (a) 7 (b) 5 (c) 3 (d) 4
14. 14.What is the smallest number you must add to 151 to make it divisible by 8? (a) 3 (b) 1 (c) 7 (d) 2
15. 15.What is the remainder when 110 is divided by 3? (a) 1 (b) 0 (c) 2
16. 16.What is the smallest number you must add to 88 to make it divisible by 3? (a) 2 (b) 1
17. 17.Which of these numbers is divisible by 3? (a) 1188 (b) 1186 (c) 1193 (d) 1189
18. 18.What is the units digit of 4^21? (a) 4 (b) 6 (c) 5 (d) 3
19. 19.What is the units digit of 9^21? (a) 1 (b) 9 (c) 7 (d) 8
20. 20.Which of these numbers is divisible by 3? (a) 2386 (b) 2388 (c) 2389 (d) 2392
21. 21.What is the remainder when 81 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
22. 22.What is the remainder when 263 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
23. 23.What is the smallest number you must add to 95 to make it divisible by 8? (a) 3 (b) 7 (c) 2 (d) 1
24. 24.What is the smallest number you must add to 225 to make it divisible by 8? (a) 1 (b) 7 (c) 6 (d) 5
25. 25.What is the smallest number you must add to 187 to make it divisible by 4? (a) 3 (b) 1 (c) 2
26. 26.Which of these numbers is divisible by 3? (a) 2558 (b) 2563 (c) 2561 (d) 2559
27. 27.What is the smallest number you must add to 82 to make it divisible by 4? (a) 1 (b) 3 (c) 2
28. 28.What is the remainder when 929 is divided by 12? (a) 6 (b) 7 (c) 4 (d) 5
29. 29.Which of these numbers is divisible by 6? (a) 3157 (b) 3171 (c) 3164 (d) 3162
30. 30.What is the remainder when 117 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 6
31. 31.Which of these numbers is divisible by 8? (a) 1887 (b) 1900 (c) 1888 (d) 1895
32. 32.What is the units digit of 2^58? (a) 4 (b) 2 (c) 6 (d) 8
33. 33.What is the smallest number you must add to 229 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 1
34. 34.Which of these numbers is divisible by 11? (a) 1322 (b) 1335 (c) 1345 (d) 1331
35. 35.Which of these numbers is divisible by 6? (a) 1207 (b) 1201 (c) 1214 (d) 1206
36. 36.What is the remainder when 828 is divided by 11? (a) 4 (b) 8 (c) 3 (d) 2
37. 37.What is the remainder when 369 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 2
38. 38.What is the remainder when 284 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
39. 39.Which of these numbers is divisible by 8? (a) 1931 (b) 1936 (c) 1945 (d) 1941
40. 40.What is the remainder when 112 is divided by 3? (a) 2 (b) 1 (c) 0

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