Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 577 · 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.Which of these numbers is divisible by 9? (a) 4662 (b) 4654 (c) 4663 (d) 4678
2. 2.What is the units digit of 4^30? (a) 4 (b) 5 (c) 6 (d) 7
3. 3.What is the remainder when 651 is divided by 12? (a) 9 (b) 2 (c) 3 (d) 4
4. 4.What is the smallest number you must add to 689 to make it divisible by 12? (a) 5 (b) 6 (c) 8 (d) 7
5. 5.Which of these numbers is divisible by 8? (a) 3443 (b) 3434 (c) 3440 (d) 3453
6. 6.What is the remainder when 580 is divided by 8? (a) 3 (b) 5 (c) 6 (d) 4
7. 7.What is the smallest number you must add to 176 to make it divisible by 3? (a) 1 (b) 2
8. 8.Which of these numbers is divisible by 4? (a) 3051 (b) 3058 (c) 3052 (d) 3053
9. 9.What is the units digit of 4^52? (a) 4 (b) 7 (c) 5 (d) 6
10. 10.What is the units digit of 8^58? (a) 8 (b) 2 (c) 4 (d) 6
11. 11.What is the units digit of 7^40? (a) 1 (b) 9 (c) 7 (d) 3
12. 12.Which of these numbers is divisible by 4? (a) 1273 (b) 1272 (c) 1277 (d) 1270
13. 13.What is the units digit of 3^19? (a) 9 (b) 3 (c) 7 (d) 1
14. 14.What is the units digit of 9^55? (a) 8 (b) 7 (c) 1 (d) 9
15. 15.Which of these numbers is divisible by 11? (a) 7304 (b) 7314 (c) 7323 (d) 7295
16. 16.What is the units digit of 2^10? (a) 8 (b) 6 (c) 2 (d) 4
17. 17.Which of these numbers is divisible by 11? (a) 2640 (b) 2639 (c) 2659 (d) 2649
18. 18.What is the smallest number you must add to 145 to make it divisible by 4? (a) 3 (b) 2 (c) 1
19. 19.What is the units digit of 3^29? (a) 9 (b) 7 (c) 3 (d) 1
20. 20.What is the units digit of 9^37? (a) 1 (b) 8 (c) 9 (d) 7
21. 21.What is the remainder when 481 is divided by 11? (a) 8 (b) 9 (c) 7 (d) 3
22. 22.What is the smallest number you must add to 164 to make it divisible by 3? (a) 1 (b) 2
23. 23.What is the smallest number you must add to 363 to make it divisible by 7? (a) 3 (b) 1 (c) 2 (d) 6
24. 24.What is the smallest number you must add to 110 to make it divisible by 3? (a) 1 (b) 2
25. 25.What is the smallest number you must add to 194 to make it divisible by 10? (a) 7 (b) 4 (c) 6 (d) 5
26. 26.What is the smallest number you must add to 380 to make it divisible by 8? (a) 6 (b) 4 (c) 5 (d) 3
27. 27.What is the remainder when 137 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 5
28. 28.Which of these numbers is divisible by 3? (a) 445 (b) 449 (c) 447 (d) 452
29. 29.What is the smallest number you must add to 338 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
30. 30.What is the units digit of 2^16? (a) 8 (b) 6 (c) 4 (d) 2
31. 31.What is the smallest number you must add to 73 to make it divisible by 4? (a) 1 (b) 3 (c) 2
32. 32.What is the smallest number you must add to 101 to make it divisible by 4? (a) 3 (b) 1 (c) 2
33. 33.Which of these numbers is divisible by 6? (a) 5238 (b) 5233 (c) 5239 (d) 5245
34. 34.Which of these numbers is divisible by 9? (a) 773 (b) 764 (c) 765 (d) 778
35. 35.What is the units digit of 7^47? (a) 7 (b) 1 (c) 9 (d) 3
36. 36.What is the units digit of 2^31? (a) 6 (b) 2 (c) 4 (d) 8
37. 37.What is the smallest number you must add to 464 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
38. 38.What is the smallest number you must add to 97 to make it divisible by 9? (a) 3 (b) 1 (c) 7 (d) 2
39. 39.What is the smallest number you must add to 238 to make it divisible by 3? (a) 1 (b) 2
40. 40.What is the remainder when 202 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2

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