Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 885 · 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 9^36? (a) 0 (b) 1 (c) 9 (d) 2
2. 2.What is the smallest number you must add to 326 to make it divisible by 7? (a) 4 (b) 3 (c) 2 (d) 5
3. 3.What is the remainder when 92 is divided by 3? (a) 1 (b) 0 (c) 2
4. 4.What is the smallest number you must add to 175 to make it divisible by 11? (a) 3 (b) 2 (c) 10 (d) 1
5. 5.What is the units digit of 7^19? (a) 3 (b) 7 (c) 9 (d) 1
6. 6.What is the smallest number you must add to 182 to make it divisible by 8? (a) 1 (b) 6 (c) 3 (d) 2
7. 7.Which of these numbers is divisible by 9? (a) 7324 (b) 7326 (c) 7334 (d) 7336
8. 8.Which of these numbers is divisible by 9? (a) 6767 (b) 6775 (c) 6768 (d) 6781
9. 9.Which of these numbers is divisible by 6? (a) 3255 (b) 3258 (c) 3260 (d) 3265
10. 10.What is the remainder when 201 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
11. 11.Which of these numbers is divisible by 6? (a) 1010 (b) 1008 (c) 1017 (d) 1006
12. 12.Which of these numbers is divisible by 9? (a) 3731 (b) 3745 (c) 3742 (d) 3735
13. 13.What is the remainder when 171 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 0
14. 14.What is the smallest number you must add to 313 to make it divisible by 10? (a) 7 (b) 6 (c) 3 (d) 8
15. 15.What is the smallest number you must add to 252 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
16. 16.What is the remainder when 129 is divided by 10? (a) 1 (b) 9 (c) 8 (d) 7
17. 17.Which of these numbers is divisible by 4? (a) 2387 (b) 2395 (c) 2389 (d) 2388
18. 18.Which of these numbers is divisible by 6? (a) 3038 (b) 3035 (c) 3025 (d) 3030
19. 19.What is the remainder when 157 is divided by 7? (a) 2 (b) 3 (c) 4 (d) 5
20. 20.Which of these numbers is divisible by 3? (a) 1592 (b) 1589 (c) 1585 (d) 1587
21. 21.What is the smallest number you must add to 469 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 4
22. 22.What is the remainder when 194 is divided by 8? (a) 1 (b) 3 (c) 6 (d) 2
23. 23.What is the smallest number you must add to 278 to make it divisible by 12? (a) 2 (b) 9 (c) 11 (d) 10
24. 24.What is the units digit of 7^25? (a) 1 (b) 3 (c) 7 (d) 9
25. 25.What is the remainder when 443 is divided by 8? (a) 3 (b) 5 (c) 2 (d) 4
26. 26.What is the units digit of 8^18? (a) 2 (b) 8 (c) 4 (d) 6
27. 27.What is the smallest number you must add to 443 to make it divisible by 10? (a) 3 (b) 6 (c) 8 (d) 7
28. 28.What is the smallest number you must add to 249 to make it divisible by 10? (a) 9 (b) 2 (c) 1 (d) 3
29. 29.What is the remainder when 169 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 2
30. 30.Which of these numbers is divisible by 6? (a) 1584 (b) 1592 (c) 1579 (d) 1587
31. 31.Which of these numbers is divisible by 3? (a) 1510 (b) 1513 (c) 1509 (d) 1508
32. 32.What is the units digit of 9^45? (a) 8 (b) 1 (c) 7 (d) 9
33. 33.What is the smallest number you must add to 880 to make it divisible by 12? (a) 7 (b) 4 (c) 8 (d) 9
34. 34.What is the units digit of 9^32? (a) 0 (b) 2 (c) 1 (d) 9
35. 35.What is the smallest number you must add to 194 to make it divisible by 4? (a) 3 (b) 1 (c) 2
36. 36.What is the remainder when 781 is divided by 10? (a) 1 (b) 0 (c) 2 (d) 9
37. 37.What is the remainder when 379 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
38. 38.What is the units digit of 7^53? (a) 9 (b) 3 (c) 1 (d) 7
39. 39.Which of these numbers is divisible by 6? (a) 4361 (b) 4363 (c) 4356 (d) 4351
40. 40.What is the units digit of 3^23? (a) 9 (b) 7 (c) 1 (d) 3

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