Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 925 · 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 178 to make it divisible by 3? (a) 2 (b) 1
2. 2.What is the remainder when 549 is divided by 11? (a) 1 (b) 8 (c) 10 (d) 9
3. 3.What is the remainder when 209 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 11? (a) 4053 (b) 4045 (c) 4036 (d) 4037
5. 5.What is the remainder when 208 is divided by 3? (a) 1 (b) 2 (c) 0
6. 6.What is the units digit of 3^43? (a) 9 (b) 1 (c) 3 (d) 7
7. 7.What is the remainder when 679 is divided by 10? (a) 7 (b) 8 (c) 9 (d) 1
8. 8.What is the units digit of 2^41? (a) 6 (b) 8 (c) 2 (d) 4
9. 9.What is the remainder when 147 is divided by 11? (a) 5 (b) 4 (c) 3 (d) 7
10. 10.What is the remainder when 79 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
11. 11.Which of these numbers is divisible by 3? (a) 310 (b) 313 (c) 309 (d) 308
12. 12.Which of these numbers is divisible by 6? (a) 5025 (b) 5028 (c) 5030 (d) 5035
13. 13.What is the units digit of 4^50? (a) 4 (b) 5 (c) 6 (d) 7
14. 14.What is the smallest number you must add to 595 to make it divisible by 8? (a) 3 (b) 5 (c) 4 (d) 6
15. 15.What is the remainder when 168 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
16. 16.Which of these numbers is divisible by 4? (a) 2528 (b) 2534 (c) 2525 (d) 2530
17. 17.What is the remainder when 597 is divided by 11? (a) 8 (b) 2 (c) 4 (d) 3
18. 18.What is the units digit of 8^47? (a) 4 (b) 8 (c) 6 (d) 2
19. 19.What is the smallest number you must add to 633 to make it divisible by 12? (a) 2 (b) 9 (c) 4 (d) 3
20. 20.What is the smallest number you must add to 142 to make it divisible by 12? (a) 1 (b) 3 (c) 2 (d) 10
21. 21.What is the remainder when 293 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
22. 22.What is the smallest number you must add to 452 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
23. 23.Which of these numbers is divisible by 4? (a) 3155 (b) 3145 (c) 3151 (d) 3148
24. 24.Which of these numbers is divisible by 8? (a) 1534 (b) 1518 (c) 1520 (d) 1521
25. 25.What is the units digit of 2^16? (a) 2 (b) 8 (c) 4 (d) 6
26. 26.What is the smallest number you must add to 496 to make it divisible by 12? (a) 7 (b) 8 (c) 9 (d) 4
27. 27.What is the units digit of 2^36? (a) 4 (b) 6 (c) 8 (d) 2
28. 28.What is the remainder when 115 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
29. 29.What is the units digit of 3^51? (a) 9 (b) 1 (c) 7 (d) 3
30. 30.Which of these numbers is divisible by 6? (a) 5020 (b) 5013 (c) 5025 (d) 5016
31. 31.What is the units digit of 2^56? (a) 6 (b) 2 (c) 4 (d) 8
32. 32.What is the remainder when 357 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
33. 33.What is the units digit of 9^11? (a) 8 (b) 1 (c) 9 (d) 7
34. 34.What is the remainder when 251 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
35. 35.What is the smallest number you must add to 107 to make it divisible by 3? (a) 2 (b) 1
36. 36.What is the units digit of 7^23? (a) 7 (b) 1 (c) 3 (d) 9
37. 37.What is the smallest number you must add to 142 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
38. 38.What is the units digit of 2^10? (a) 4 (b) 6 (c) 8 (d) 2
39. 39.What is the smallest number you must add to 596 to make it divisible by 9? (a) 6 (b) 2 (c) 8 (d) 7
40. 40.What is the units digit of 3^59? (a) 9 (b) 3 (c) 7 (d) 1

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