Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 141 · 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 59 is divided by 3? (a) 0 (b) 2 (c) 1
2. 2.What is the smallest number you must add to 202 to make it divisible by 8? (a) 6 (b) 2 (c) 7 (d) 5
3. 3.What is the units digit of 8^10? (a) 4 (b) 8 (c) 6 (d) 2
4. 4.What is the smallest number you must add to 353 to make it divisible by 9? (a) 2 (b) 8 (c) 6 (d) 7
5. 5.What is the units digit of 7^44? (a) 9 (b) 7 (c) 3 (d) 1
6. 6.What is the remainder when 383 is divided by 12? (a) 9 (b) 11 (c) 10 (d) 1
7. 7.What is the smallest number you must add to 155 to make it divisible by 7? (a) 1 (b) 5 (c) 6 (d) 4
8. 8.What is the remainder when 106 is divided by 3? (a) 2 (b) 1 (c) 0
9. 9.What is the smallest number you must add to 639 to make it divisible by 10? (a) 3 (b) 1 (c) 2 (d) 9
10. 10.What is the smallest number you must add to 495 to make it divisible by 10? (a) 5 (b) 7 (c) 6 (d) 4
11. 11.What is the units digit of 9^45? (a) 8 (b) 9 (c) 7 (d) 1
12. 12.Which of these numbers is divisible by 6? (a) 3552 (b) 3553 (c) 3559 (d) 3551
13. 13.Which of these numbers is divisible by 4? (a) 3085 (b) 3080 (c) 3081 (d) 3077
14. 14.What is the remainder when 275 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 2
15. 15.What is the remainder when 200 is divided by 3? (a) 0 (b) 1 (c) 2
16. 16.Which of these numbers is divisible by 9? (a) 2241 (b) 2247 (c) 2235 (d) 2251
17. 17.What is the units digit of 4^13? (a) 4 (b) 6 (c) 3 (d) 5
18. 18.What is the smallest number you must add to 243 to make it divisible by 7? (a) 3 (b) 5 (c) 1 (d) 2
19. 19.What is the units digit of 3^32? (a) 9 (b) 7 (c) 3 (d) 1
20. 20.Which of these numbers is divisible by 8? (a) 3593 (b) 3584 (c) 3577 (d) 3585
21. 21.What is the units digit of 7^29? (a) 3 (b) 9 (c) 7 (d) 1
22. 22.What is the smallest number you must add to 262 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
23. 23.What is the units digit of 4^52? (a) 7 (b) 4 (c) 6 (d) 5
24. 24.What is the units digit of 9^10? (a) 1 (b) 0 (c) 2 (d) 9
25. 25.What is the units digit of 7^33? (a) 7 (b) 1 (c) 9 (d) 3
26. 26.Which of these numbers is divisible by 8? (a) 2395 (b) 2392 (c) 2391 (d) 2401
27. 27.What is the remainder when 44 is divided by 3? (a) 0 (b) 1 (c) 2
28. 28.Which of these numbers is divisible by 8? (a) 679 (b) 672 (c) 667 (d) 687
29. 29.Which of these numbers is divisible by 4? (a) 2978 (b) 2976 (c) 2973 (d) 2982
30. 30.What is the smallest number you must add to 393 to make it divisible by 7? (a) 4 (b) 1 (c) 5 (d) 6
31. 31.Which of these numbers is divisible by 9? (a) 6804 (b) 6815 (c) 6800 (d) 6805
32. 32.Which of these numbers is divisible by 4? (a) 434 (b) 438 (c) 431 (d) 432
33. 33.What is the remainder when 390 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 2
34. 34.What is the units digit of 9^13? (a) 8 (b) 1 (c) 9 (d) 7
35. 35.What is the remainder when 235 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
36. 36.Which of these numbers is divisible by 9? (a) 5924 (b) 5921 (c) 5913 (d) 5912
37. 37.What is the remainder when 309 is divided by 12? (a) 3 (b) 8 (c) 10 (d) 9
38. 38.What is the remainder when 185 is divided by 3? (a) 1 (b) 2 (c) 0
39. 39.What is the units digit of 4^49? (a) 6 (b) 3 (c) 4 (d) 5
40. 40.What is the remainder when 91 is divided by 3? (a) 2 (b) 0 (c) 1

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