Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 940 · 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 remainder when 675 is divided by 12? (a) 3 (b) 9 (c) 4 (d) 2
2. 2.What is the units digit of 2^42? (a) 4 (b) 6 (c) 2 (d) 8
3. 3.What is the remainder when 269 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
4. 4.What is the units digit of 8^21? (a) 6 (b) 2 (c) 4 (d) 8
5. 5.Which of these numbers is divisible by 6? (a) 4963 (b) 4956 (c) 4961 (d) 4951
6. 6.What is the remainder when 263 is divided by 8? (a) 5 (b) 6 (c) 1 (d) 7
7. 7.What is the remainder when 229 is divided by 3? (a) 1 (b) 0 (c) 2
8. 8.What is the smallest number you must add to 186 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
9. 9.What is the units digit of 8^26? (a) 6 (b) 4 (c) 2 (d) 8
10. 10.What is the remainder when 601 is divided by 8? (a) 2 (b) 0 (c) 7 (d) 1
11. 11.What is the smallest number you must add to 138 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
12. 12.What is the remainder when 190 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
13. 13.What is the units digit of 7^12? (a) 9 (b) 3 (c) 7 (d) 1
14. 14.What is the smallest number you must add to 241 to make it divisible by 3? (a) 1 (b) 2
15. 15.What is the remainder when 146 is divided by 3? (a) 0 (b) 1 (c) 2
16. 16.Which of these numbers is divisible by 8? (a) 1137 (b) 1130 (c) 1128 (d) 1122
17. 17.Which of these numbers is divisible by 3? (a) 1772 (b) 1768 (c) 1770 (d) 1774
18. 18.What is the units digit of 8^53? (a) 2 (b) 6 (c) 8 (d) 4
19. 19.What is the smallest number you must add to 189 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 2
20. 20.What is the remainder when 126 is divided by 11? (a) 6 (b) 5 (c) 4 (d) 7
21. 21.What is the smallest number you must add to 416 to make it divisible by 9? (a) 6 (b) 2 (c) 8 (d) 7
22. 22.What is the remainder when 471 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 2
23. 23.What is the smallest number you must add to 172 to make it divisible by 7? (a) 3 (b) 2 (c) 5 (d) 4
24. 24.What is the units digit of 3^13? (a) 1 (b) 7 (c) 3 (d) 9
25. 25.What is the units digit of 9^28? (a) 2 (b) 9 (c) 1 (d) 0
26. 26.What is the remainder when 187 is divided by 6? (a) 2 (b) 5 (c) 1 (d) 0
27. 27.Which of these numbers is divisible by 6? (a) 1661 (b) 1662 (c) 1664 (d) 1671
28. 28.What is the smallest number you must add to 151 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 4
29. 29.What is the remainder when 128 is divided by 10? (a) 8 (b) 9 (c) 7 (d) 2
30. 30.Which of these numbers is divisible by 6? (a) 2656 (b) 2658 (c) 2666 (d) 2663
31. 31.What is the units digit of 4^24? (a) 5 (b) 7 (c) 4 (d) 6
32. 32.What is the units digit of 2^22? (a) 4 (b) 8 (c) 6 (d) 2
33. 33.What is the smallest number you must add to 260 to make it divisible by 12? (a) 8 (b) 4 (c) 3 (d) 5
34. 34.What is the units digit of 4^17? (a) 6 (b) 5 (c) 3 (d) 4
35. 35.What is the remainder when 589 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 6
36. 36.What is the smallest number you must add to 173 to make it divisible by 8? (a) 5 (b) 2 (c) 4 (d) 3
37. 37.Which of these numbers is divisible by 6? (a) 3438 (b) 3441 (c) 3437 (d) 3449
38. 38.What is the units digit of 3^18? (a) 9 (b) 3 (c) 7 (d) 1
39. 39.What is the units digit of 2^43? (a) 8 (b) 4 (c) 6 (d) 2
40. 40.Which of these numbers is divisible by 9? (a) 5488 (b) 5493 (c) 5481 (d) 5475

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