Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 677 · 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 4? (a) 2137 (b) 2143 (c) 2134 (d) 2136
2. 2.What is the units digit of 9^43? (a) 7 (b) 1 (c) 8 (d) 9
3. 3.What is the remainder when 157 is divided by 6? (a) 0 (b) 2 (c) 1 (d) 5
4. 4.What is the remainder when 170 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 1
5. 5.What is the units digit of 2^48? (a) 6 (b) 8 (c) 4 (d) 2
6. 6.What is the units digit of 2^14? (a) 2 (b) 8 (c) 6 (d) 4
7. 7.What is the smallest number you must add to 119 to make it divisible by 3? (a) 2 (b) 1
8. 8.What is the smallest number you must add to 301 to make it divisible by 12? (a) 11 (b) 9 (c) 1 (d) 10
9. 9.What is the remainder when 79 is divided by 3? (a) 1 (b) 2 (c) 0
10. 10.What is the units digit of 8^56? (a) 4 (b) 6 (c) 2 (d) 8
11. 11.Which of these numbers is divisible by 8? (a) 4608 (b) 4618 (c) 4615 (d) 4607
12. 12.Which of these numbers is divisible by 9? (a) 2438 (b) 2422 (c) 2430 (d) 2440
13. 13.What is the units digit of 3^45? (a) 1 (b) 9 (c) 3 (d) 7
14. 14.What is the units digit of 7^13? (a) 7 (b) 9 (c) 1 (d) 3
15. 15.Which of these numbers is divisible by 9? (a) 6662 (b) 6660 (c) 6652 (d) 6673
16. 16.Which of these numbers is divisible by 6? (a) 4731 (b) 4722 (c) 4720 (d) 4723
17. 17.Which of these numbers is divisible by 9? (a) 954 (b) 950 (c) 959 (d) 965
18. 18.What is the smallest number you must add to 296 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
19. 19.What is the smallest number you must add to 446 to make it divisible by 9? (a) 3 (b) 5 (c) 4 (d) 6
20. 20.Which of these numbers is divisible by 3? (a) 1552 (b) 1554 (c) 1556 (d) 1558
21. 21.What is the units digit of 2^13? (a) 4 (b) 6 (c) 8 (d) 2
22. 22.Which of these numbers is divisible by 4? (a) 2683 (b) 2687 (c) 2690 (d) 2684
23. 23.What is the smallest number you must add to 735 to make it divisible by 10? (a) 5 (b) 7 (c) 4 (d) 6
24. 24.What is the units digit of 4^59? (a) 3 (b) 6 (c) 5 (d) 4
25. 25.Which of these numbers is divisible by 4? (a) 2633 (b) 2625 (c) 2628 (d) 2629
26. 26.What is the units digit of 9^27? (a) 7 (b) 8 (c) 9 (d) 1
27. 27.What is the units digit of 2^44? (a) 8 (b) 4 (c) 6 (d) 2
28. 28.What is the smallest number you must add to 82 to make it divisible by 4? (a) 3 (b) 1 (c) 2
29. 29.What is the remainder when 43 is divided by 3? (a) 1 (b) 2 (c) 0
30. 30.Which of these numbers is divisible by 8? (a) 3105 (b) 3091 (c) 3096 (d) 3103
31. 31.What is the remainder when 201 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
32. 32.Which of these numbers is divisible by 4? (a) 2250 (b) 2245 (c) 2243 (d) 2244
33. 33.What is the remainder when 95 is divided by 3? (a) 2 (b) 1 (c) 0
34. 34.What is the remainder when 137 is divided by 11? (a) 7 (b) 5 (c) 6 (d) 4
35. 35.What is the remainder when 158 is divided by 3? (a) 0 (b) 2 (c) 1
36. 36.Which of these numbers is divisible by 11? (a) 8078 (b) 8073 (c) 8061 (d) 8063
37. 37.What is the remainder when 151 is divided by 3? (a) 2 (b) 0 (c) 1
38. 38.What is the smallest number you must add to 172 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
39. 39.What is the units digit of 8^12? (a) 8 (b) 4 (c) 2 (d) 6
40. 40.Which of these numbers is divisible by 11? (a) 2926 (b) 2933 (c) 2924 (d) 2943

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