Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 626 · 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 158 to make it divisible by 3? (a) 1 (b) 2
2. 2.What is the remainder when 278 is divided by 8? (a) 7 (b) 6 (c) 2 (d) 5
3. 3.Which of these numbers is divisible by 9? (a) 5531 (b) 5517 (c) 5522 (d) 5509
4. 4.Which of these numbers is divisible by 3? (a) 334 (b) 337 (c) 333 (d) 331
5. 5.What is the smallest number you must add to 726 to make it divisible by 10? (a) 3 (b) 6 (c) 5 (d) 4
6. 6.What is the remainder when 313 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
7. 7.Which of these numbers is divisible by 9? (a) 3841 (b) 3843 (c) 3860 (d) 3846
8. 8.What is the units digit of 3^31? (a) 1 (b) 7 (c) 3 (d) 9
9. 9.What is the units digit of 8^34? (a) 4 (b) 2 (c) 8 (d) 6
10. 10.What is the smallest number you must add to 177 to make it divisible by 4? (a) 3 (b) 1 (c) 2
11. 11.What is the units digit of 2^37? (a) 2 (b) 4 (c) 8 (d) 6
12. 12.Which of these numbers is divisible by 6? (a) 2354 (b) 2346 (c) 2347 (d) 2344
13. 13.What is the smallest number you must add to 457 to make it divisible by 9? (a) 1 (b) 3 (c) 2 (d) 7
14. 14.Which of these numbers is divisible by 9? (a) 6530 (b) 6542 (c) 6517 (d) 6525
15. 15.What is the smallest number you must add to 181 to make it divisible by 4? (a) 3 (b) 2 (c) 1
16. 16.What is the remainder when 343 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
17. 17.What is the remainder when 850 is divided by 11? (a) 8 (b) 2 (c) 3 (d) 4
18. 18.Which of these numbers is divisible by 3? (a) 292 (b) 298 (c) 294 (d) 295
19. 19.What is the remainder when 155 is divided by 6? (a) 5 (b) 1 (c) 3 (d) 4
20. 20.What is the smallest number you must add to 71 to make it divisible by 7? (a) 6 (b) 5 (c) 1 (d) 4
21. 21.Which of these numbers is divisible by 4? (a) 321 (b) 316 (c) 317 (d) 315
22. 22.Which of these numbers is divisible by 11? (a) 4879 (b) 4901 (c) 4884 (d) 4892
23. 23.What is the units digit of 2^46? (a) 6 (b) 2 (c) 4 (d) 8
24. 24.Which of these numbers is divisible by 9? (a) 2291 (b) 2279 (c) 2286 (d) 2300
25. 25.Which of these numbers is divisible by 9? (a) 6129 (b) 6141 (c) 6130 (d) 6124
26. 26.What is the smallest number you must add to 146 to make it divisible by 3? (a) 1 (b) 2
27. 27.What is the remainder when 311 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
28. 28.What is the units digit of 3^10? (a) 3 (b) 9 (c) 7 (d) 1
29. 29.What is the smallest number you must add to 167 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
30. 30.Which of these numbers is divisible by 8? (a) 1636 (b) 1632 (c) 1625 (d) 1644
31. 31.What is the remainder when 325 is divided by 6? (a) 5 (b) 1 (c) 0 (d) 2
32. 32.What is the remainder when 696 is divided by 9? (a) 4 (b) 2 (c) 6 (d) 3
33. 33.What is the units digit of 4^10? (a) 4 (b) 6 (c) 7 (d) 5
34. 34.What is the units digit of 3^40? (a) 3 (b) 7 (c) 9 (d) 1
35. 35.What is the units digit of 2^43? (a) 8 (b) 4 (c) 2 (d) 6
36. 36.Which of these numbers is divisible by 11? (a) 2233 (b) 2239 (c) 2230 (d) 2250
37. 37.Which of these numbers is divisible by 9? (a) 3588 (b) 3591 (c) 3607 (d) 3592
38. 38.What is the remainder when 92 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
39. 39.What is the smallest number you must add to 212 to make it divisible by 3? (a) 2 (b) 1
40. 40.What is the smallest number you must add to 320 to make it divisible by 12? (a) 4 (b) 5 (c) 3 (d) 8

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