Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 753 · 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 50 to make it divisible by 4? (a) 2 (b) 3 (c) 1
2. 2.What is the remainder when 96 is divided by 5? (a) 4 (b) 0 (c) 2 (d) 1
3. 3.What is the smallest number you must add to 281 to make it divisible by 7? (a) 6 (b) 1 (c) 4 (d) 5
4. 4.Which of these numbers is divisible by 8? (a) 649 (b) 640 (c) 641 (d) 633
5. 5.What is the remainder when 159 is divided by 10? (a) 9 (b) 1 (c) 8 (d) 7
6. 6.Which of these numbers is divisible by 4? (a) 3275 (b) 3271 (c) 3272 (d) 3278
7. 7.What is the smallest number you must add to 94 to make it divisible by 3? (a) 1 (b) 2
8. 8.Which of these numbers is divisible by 6? (a) 3909 (b) 3904 (c) 3917 (d) 3906
9. 9.What is the units digit of 2^44? (a) 2 (b) 4 (c) 8 (d) 6
10. 10.Which of these numbers is divisible by 6? (a) 1976 (b) 1967 (c) 1968 (d) 1971
11. 11.What is the remainder when 149 is divided by 3? (a) 0 (b) 1 (c) 2
12. 12.What is the units digit of 4^20? (a) 7 (b) 6 (c) 5 (d) 4
13. 13.Which of these numbers is divisible by 3? (a) 1771 (b) 1765 (c) 1768 (d) 1767
14. 14.What is the smallest number you must add to 173 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
15. 15.Which of these numbers is divisible by 4? (a) 2169 (b) 2167 (c) 2175 (d) 2168
16. 16.What is the units digit of 2^21? (a) 6 (b) 8 (c) 2 (d) 4
17. 17.What is the units digit of 3^38? (a) 3 (b) 7 (c) 9 (d) 1
18. 18.What is the remainder when 514 is divided by 12? (a) 9 (b) 2 (c) 11 (d) 10
19. 19.What is the remainder when 290 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 2
20. 20.Which of these numbers is divisible by 11? (a) 5012 (b) 5019 (c) 5002 (d) 5005
21. 21.Which of these numbers is divisible by 9? (a) 7295 (b) 7281 (c) 7282 (d) 7279
22. 22.What is the smallest number you must add to 341 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
23. 23.What is the units digit of 7^14? (a) 1 (b) 7 (c) 9 (d) 3
24. 24.What is the smallest number you must add to 218 to make it divisible by 3? (a) 2 (b) 1
25. 25.What is the remainder when 576 is divided by 10? (a) 7 (b) 6 (c) 5 (d) 4
26. 26.Which of these numbers is divisible by 4? (a) 1725 (b) 1735 (c) 1731 (d) 1728
27. 27.What is the units digit of 9^17? (a) 7 (b) 9 (c) 8 (d) 1
28. 28.What is the units digit of 9^52? (a) 9 (b) 0 (c) 2 (d) 1
29. 29.What is the smallest number you must add to 539 to make it divisible by 9? (a) 3 (b) 2 (c) 1 (d) 8
30. 30.Which of these numbers is divisible by 3? (a) 2071 (b) 2068 (c) 2066 (d) 2067
31. 31.Which of these numbers is divisible by 9? (a) 5347 (b) 5358 (c) 5343 (d) 5346
32. 32.What is the remainder when 173 is divided by 10? (a) 7 (b) 2 (c) 3 (d) 4
33. 33.What is the remainder when 71 is divided by 7? (a) 1 (b) 0 (c) 2 (d) 6
34. 34.What is the units digit of 4^33? (a) 6 (b) 5 (c) 4 (d) 3
35. 35.What is the units digit of 3^49? (a) 7 (b) 9 (c) 1 (d) 3
36. 36.What is the remainder when 258 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
37. 37.What is the smallest number you must add to 177 to make it divisible by 11? (a) 10 (b) 1 (c) 9 (d) 8
38. 38.What is the units digit of 3^12? (a) 1 (b) 9 (c) 7 (d) 3
39. 39.What is the units digit of 2^32? (a) 8 (b) 4 (c) 6 (d) 2
40. 40.Which of these numbers is divisible by 9? (a) 2583 (b) 2594 (c) 2584 (d) 2580

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