Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 54 · 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.Which of these numbers is divisible by 4? (a) 362 (b) 358 (c) 365 (d) 360
2. 2.What is the remainder when 632 is divided by 9? (a) 3 (b) 2 (c) 1 (d) 7
3. 3.What is the remainder when 88 is divided by 3? (a) 2 (b) 0 (c) 1
4. 4.What is the units digit of 9^20? (a) 0 (b) 2 (c) 1 (d) 9
5. 5.What is the units digit of 9^50? (a) 1 (b) 2 (c) 0 (d) 9
6. 6.What is the remainder when 368 is divided by 10? (a) 2 (b) 8 (c) 7 (d) 9
7. 7.What is the smallest number you must add to 279 to make it divisible by 11? (a) 8 (b) 6 (c) 7 (d) 4
8. 8.What is the units digit of 9^52? (a) 9 (b) 0 (c) 1 (d) 2
9. 9.What is the remainder when 157 is divided by 9? (a) 3 (b) 4 (c) 5 (d) 6
10. 10.What is the units digit of 2^50? (a) 8 (b) 2 (c) 4 (d) 6
11. 11.What is the remainder when 310 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 5
12. 12.What is the smallest number you must add to 326 to make it divisible by 12? (a) 11 (b) 2 (c) 9 (d) 10
13. 13.Which of these numbers is divisible by 6? (a) 3013 (b) 3007 (c) 3023 (d) 3012
14. 14.Which of these numbers is divisible by 3? (a) 196 (b) 195 (c) 194 (d) 200
15. 15.What is the units digit of 4^14? (a) 6 (b) 5 (c) 7 (d) 4
16. 16.What is the remainder when 162 is divided by 10? (a) 3 (b) 1 (c) 2 (d) 8
17. 17.Which of these numbers is divisible by 8? (a) 6656 (b) 6662 (c) 6651 (d) 6671
18. 18.Which of these numbers is divisible by 9? (a) 1221 (b) 1206 (c) 1210 (d) 1201
19. 19.What is the units digit of 3^16? (a) 7 (b) 9 (c) 3 (d) 1
20. 20.What is the smallest number you must add to 238 to make it divisible by 3? (a) 2 (b) 1
21. 21.What is the smallest number you must add to 421 to make it divisible by 10? (a) 8 (b) 9 (c) 7 (d) 1
22. 22.What is the units digit of 2^25? (a) 6 (b) 2 (c) 4 (d) 8
23. 23.What is the smallest number you must add to 541 to make it divisible by 8? (a) 5 (b) 3 (c) 2 (d) 4
24. 24.What is the remainder when 428 is divided by 10? (a) 7 (b) 2 (c) 8 (d) 9
25. 25.What is the remainder when 389 is divided by 9? (a) 1 (b) 7 (c) 2 (d) 3
26. 26.Which of these numbers is divisible by 6? (a) 2185 (b) 2183 (c) 2178 (d) 2173
27. 27.What is the units digit of 9^49? (a) 8 (b) 7 (c) 9 (d) 1
28. 28.What is the remainder when 539 is divided by 8? (a) 2 (b) 5 (c) 4 (d) 3
29. 29.What is the units digit of 3^31? (a) 7 (b) 1 (c) 9 (d) 3
30. 30.What is the smallest number you must add to 69 to make it divisible by 4? (a) 2 (b) 3 (c) 1
31. 31.What is the units digit of 9^34? (a) 9 (b) 1 (c) 0 (d) 2
32. 32.Which of these numbers is divisible by 11? (a) 3090 (b) 3072 (c) 3098 (d) 3080
33. 33.What is the units digit of 8^46? (a) 2 (b) 6 (c) 4 (d) 8
34. 34.Which of these numbers is divisible by 3? (a) 271 (b) 278 (c) 275 (d) 273
35. 35.What is the remainder when 588 is divided by 11? (a) 6 (b) 7 (c) 5 (d) 4
36. 36.What is the units digit of 9^17? (a) 1 (b) 7 (c) 8 (d) 9
37. 37.What is the remainder when 75 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
38. 38.What is the remainder when 273 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
39. 39.What is the remainder when 319 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
40. 40.What is the units digit of 7^50? (a) 1 (b) 3 (c) 9 (d) 7

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