Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 355 · 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 361 is divided by 12? (a) 2 (b) 11 (c) 0 (d) 1
2. 2.What is the remainder when 211 is divided by 3? (a) 2 (b) 0 (c) 1
3. 3.What is the smallest number you must add to 602 to make it divisible by 12? (a) 9 (b) 11 (c) 10 (d) 2
4. 4.What is the units digit of 7^41? (a) 9 (b) 7 (c) 3 (d) 1
5. 5.Which of these numbers is divisible by 11? (a) 5173 (b) 5156 (c) 5159 (d) 5164
6. 6.Which of these numbers is divisible by 4? (a) 3406 (b) 3404 (c) 3409 (d) 3402
7. 7.Which of these numbers is divisible by 4? (a) 3440 (b) 3441 (c) 3439 (d) 3445
8. 8.What is the units digit of 7^24? (a) 3 (b) 7 (c) 9 (d) 1
9. 9.What is the remainder when 442 is divided by 12? (a) 2 (b) 11 (c) 10 (d) 9
10. 10.What is the smallest number you must add to 480 to make it divisible by 11? (a) 5 (b) 3 (c) 4 (d) 7
11. 11.What is the smallest number you must add to 526 to make it divisible by 9? (a) 4 (b) 7 (c) 5 (d) 6
12. 12.Which of these numbers is divisible by 8? (a) 2140 (b) 2128 (c) 2132 (d) 2122
13. 13.What is the remainder when 559 is divided by 9? (a) 2 (b) 0 (c) 8 (d) 1
14. 14.What is the remainder when 237 is divided by 8? (a) 3 (b) 5 (c) 6 (d) 4
15. 15.What is the remainder when 218 is divided by 9? (a) 2 (b) 3 (c) 7 (d) 1
16. 16.What is the units digit of 2^11? (a) 6 (b) 2 (c) 4 (d) 8
17. 17.What is the smallest number you must add to 226 to make it divisible by 3? (a) 1 (b) 2
18. 18.Which of these numbers is divisible by 8? (a) 1983 (b) 1984 (c) 1997 (d) 1988
19. 19.What is the remainder when 117 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
20. 20.What is the remainder when 591 is divided by 8? (a) 7 (b) 5 (c) 1 (d) 6
21. 21.Which of these numbers is divisible by 3? (a) 332 (b) 334 (c) 329 (d) 330
22. 22.Which of these numbers is divisible by 9? (a) 2367 (b) 2364 (c) 2371 (d) 2381
23. 23.What is the smallest number you must add to 195 to make it divisible by 7? (a) 2 (b) 1 (c) 6 (d) 3
24. 24.What is the smallest number you must add to 306 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
25. 25.What is the smallest number you must add to 221 to make it divisible by 4? (a) 1 (b) 2 (c) 3
26. 26.What is the units digit of 3^11? (a) 7 (b) 1 (c) 9 (d) 3
27. 27.Which of these numbers is divisible by 6? (a) 5229 (b) 5235 (c) 5226 (d) 5224
28. 28.What is the smallest number you must add to 137 to make it divisible by 7? (a) 2 (b) 3 (c) 4 (d) 5
29. 29.What is the smallest number you must add to 101 to make it divisible by 3? (a) 2 (b) 1
30. 30.What is the remainder when 106 is divided by 3? (a) 2 (b) 0 (c) 1
31. 31.Which of these numbers is divisible by 6? (a) 3720 (b) 3724 (c) 3718 (d) 3727
32. 32.Which of these numbers is divisible by 8? (a) 366 (b) 371 (c) 381 (d) 368
33. 33.What is the units digit of 3^17? (a) 1 (b) 9 (c) 3 (d) 7
34. 34.What is the remainder when 193 is divided by 3? (a) 1 (b) 0 (c) 2
35. 35.Which of these numbers is divisible by 3? (a) 2460 (b) 2464 (c) 2459 (d) 2461
36. 36.What is the units digit of 4^32? (a) 5 (b) 7 (c) 6 (d) 4
37. 37.What is the remainder when 278 is divided by 12? (a) 3 (b) 2 (c) 1 (d) 10
38. 38.What is the remainder when 151 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 3
39. 39.Which of these numbers is divisible by 3? (a) 298 (b) 293 (c) 296 (d) 294
40. 40.What is the remainder when 173 is divided by 3? (a) 2 (b) 0 (c) 1

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