Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 171 · 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 4^21? (a) 6 (b) 5 (c) 3 (d) 4
2. 2.What is the smallest number you must add to 292 to make it divisible by 11? (a) 4 (b) 7 (c) 6 (d) 5
3. 3.What is the remainder when 235 is divided by 3? (a) 1 (b) 0 (c) 2
4. 4.What is the remainder when 158 is divided by 3? (a) 2 (b) 1 (c) 0
5. 5.What is the units digit of 9^21? (a) 8 (b) 9 (c) 1 (d) 7
6. 6.Which of these numbers is divisible by 3? (a) 1214 (b) 1207 (c) 1209 (d) 1211
7. 7.What is the smallest number you must add to 133 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
8. 8.What is the smallest number you must add to 233 to make it divisible by 10? (a) 3 (b) 8 (c) 6 (d) 7
9. 9.What is the remainder when 218 is divided by 7? (a) 0 (b) 6 (c) 2 (d) 1
10. 10.What is the units digit of 4^10? (a) 7 (b) 5 (c) 6 (d) 4
11. 11.What is the remainder when 525 is divided by 11? (a) 7 (b) 3 (c) 9 (d) 8
12. 12.What is the units digit of 4^58? (a) 4 (b) 6 (c) 7 (d) 5
13. 13.Which of these numbers is divisible by 4? (a) 2425 (b) 2424 (c) 2421 (d) 2429
14. 14.What is the smallest number you must add to 282 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
15. 15.What is the remainder when 171 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
16. 16.Which of these numbers is divisible by 11? (a) 6120 (b) 6130 (c) 6148 (d) 6127
17. 17.What is the remainder when 656 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
18. 18.What is the smallest number you must add to 106 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
19. 19.Which of these numbers is divisible by 9? (a) 3437 (b) 3412 (c) 3427 (d) 3420
20. 20.What is the remainder when 106 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
21. 21.What is the remainder when 379 is divided by 9? (a) 2 (b) 0 (c) 8 (d) 1
22. 22.Which of these numbers is divisible by 3? (a) 2148 (b) 2146 (c) 2153 (d) 2149
23. 23.Which of these numbers is divisible by 9? (a) 3169 (b) 3162 (c) 3180 (d) 3168
24. 24.What is the smallest number you must add to 89 to make it divisible by 3? (a) 2 (b) 1
25. 25.What is the remainder when 423 is divided by 8? (a) 5 (b) 1 (c) 7 (d) 6
26. 26.Which of these numbers is divisible by 6? (a) 4663 (b) 4658 (c) 4671 (d) 4662
27. 27.What is the smallest number you must add to 174 to make it divisible by 4? (a) 3 (b) 1 (c) 2
28. 28.Which of these numbers is divisible by 8? (a) 2024 (b) 2029 (c) 2035 (d) 2022
29. 29.Which of these numbers is divisible by 11? (a) 7483 (b) 7475 (c) 7465 (d) 7469
30. 30.What is the remainder when 173 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
31. 31.What is the remainder when 287 is divided by 11? (a) 2 (b) 0 (c) 1 (d) 10
32. 32.Which of these numbers is divisible by 6? (a) 1507 (b) 1501 (c) 1506 (d) 1515
33. 33.What is the units digit of 8^51? (a) 4 (b) 6 (c) 8 (d) 2
34. 34.Which of these numbers is divisible by 3? (a) 2290 (b) 2289 (c) 2294 (d) 2288
35. 35.What is the smallest number you must add to 64 to make it divisible by 3? (a) 2 (b) 1
36. 36.What is the remainder when 546 is divided by 8? (a) 2 (b) 3 (c) 1 (d) 6
37. 37.What is the remainder when 322 is divided by 9? (a) 7 (b) 2 (c) 8 (d) 6
38. 38.What is the smallest number you must add to 81 to make it divisible by 4? (a) 1 (b) 3 (c) 2
39. 39.What is the units digit of 7^29? (a) 1 (b) 7 (c) 9 (d) 3
40. 40.What is the smallest number you must add to 276 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1

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