Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 359 · 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 units digit of 8^47? (a) 6 (b) 4 (c) 8 (d) 2
2. 2.Which of these numbers is divisible by 3? (a) 2500 (b) 2498 (c) 2495 (d) 2496
3. 3.What is the smallest number you must add to 290 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 2
4. 4.What is the remainder when 90 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
5. 5.What is the units digit of 4^26? (a) 5 (b) 4 (c) 6 (d) 7
6. 6.What is the units digit of 2^54? (a) 4 (b) 2 (c) 8 (d) 6
7. 7.What is the units digit of 8^33? (a) 8 (b) 6 (c) 4 (d) 2
8. 8.What is the remainder when 178 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
9. 9.Which of these numbers is divisible by 3? (a) 1468 (b) 1475 (c) 1472 (d) 1470
10. 10.What is the smallest number you must add to 314 to make it divisible by 4? (a) 3 (b) 2 (c) 1
11. 11.Which of these numbers is divisible by 9? (a) 3676 (b) 3684 (c) 3670 (d) 3672
12. 12.Which of these numbers is divisible by 8? (a) 5504 (b) 5519 (c) 5501 (d) 5505
13. 13.Which of these numbers is divisible by 9? (a) 1124 (b) 1114 (c) 1116 (d) 1133
14. 14.Which of these numbers is divisible by 3? (a) 1755 (b) 1754 (c) 1756 (d) 1759
15. 15.What is the units digit of 2^55? (a) 8 (b) 2 (c) 4 (d) 6
16. 16.What is the remainder when 362 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
17. 17.Which of these numbers is divisible by 8? (a) 5554 (b) 5566 (c) 5560 (d) 5575
18. 18.Which of these numbers is divisible by 6? (a) 801 (b) 798 (c) 796 (d) 806
19. 19.What is the remainder when 656 is divided by 12? (a) 7 (b) 9 (c) 4 (d) 8
20. 20.What is the remainder when 632 is divided by 9? (a) 7 (b) 3 (c) 2 (d) 1
21. 21.What is the remainder when 802 is divided by 11? (a) 8 (b) 10 (c) 1 (d) 9
22. 22.Which of these numbers is divisible by 3? (a) 1512 (b) 1517 (c) 1513 (d) 1510
23. 23.What is the units digit of 2^17? (a) 4 (b) 2 (c) 6 (d) 8
24. 24.What is the smallest number you must add to 377 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 5
25. 25.What is the smallest number you must add to 101 to make it divisible by 4? (a) 2 (b) 3 (c) 1
26. 26.Which of these numbers is divisible by 3? (a) 1311 (b) 1315 (c) 1309 (d) 1313
27. 27.What is the units digit of 9^20? (a) 0 (b) 1 (c) 2 (d) 9
28. 28.Which of these numbers is divisible by 6? (a) 1631 (b) 1637 (c) 1639 (d) 1632
29. 29.What is the units digit of 4^31? (a) 6 (b) 3 (c) 5 (d) 4
30. 30.What is the smallest number you must add to 174 to make it divisible by 4? (a) 2 (b) 3 (c) 1
31. 31.What is the units digit of 4^24? (a) 7 (b) 4 (c) 5 (d) 6
32. 32.Which of these numbers is divisible by 11? (a) 3528 (b) 3510 (c) 3535 (d) 3520
33. 33.What is the units digit of 7^40? (a) 7 (b) 1 (c) 3 (d) 9
34. 34.What is the remainder when 344 is divided by 9? (a) 1 (b) 7 (c) 2 (d) 3
35. 35.What is the smallest number you must add to 388 to make it divisible by 10? (a) 8 (b) 3 (c) 1 (d) 2
36. 36.What is the units digit of 4^15? (a) 4 (b) 6 (c) 5 (d) 3
37. 37.Which of these numbers is divisible by 11? (a) 2308 (b) 2318 (c) 2310 (d) 2327
38. 38.What is the remainder when 154 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
39. 39.Which of these numbers is divisible by 4? (a) 3074 (b) 3083 (c) 3076 (d) 3079
40. 40.What is the smallest number you must add to 98 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3

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