Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 953 · 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 133 is divided by 11? (a) 10 (b) 2 (c) 0 (d) 1
2. 2.Which of these numbers is divisible by 9? (a) 7028 (b) 7020 (c) 7030 (d) 7018
3. 3.What is the remainder when 643 is divided by 11? (a) 7 (b) 6 (c) 5 (d) 4
4. 4.What is the units digit of 7^57? (a) 1 (b) 9 (c) 3 (d) 7
5. 5.Which of these numbers is divisible by 8? (a) 1100 (b) 1118 (c) 1104 (d) 1106
6. 6.What is the smallest number you must add to 284 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
7. 7.What is the remainder when 133 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
8. 8.What is the units digit of 2^43? (a) 4 (b) 2 (c) 6 (d) 8
9. 9.What is the remainder when 71 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
10. 10.What is the smallest number you must add to 556 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
11. 11.What is the smallest number you must add to 165 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
12. 12.What is the remainder when 128 is divided by 3? (a) 1 (b) 2 (c) 0
13. 13.What is the units digit of 4^11? (a) 3 (b) 5 (c) 6 (d) 4
14. 14.What is the remainder when 314 is divided by 8? (a) 1 (b) 3 (c) 6 (d) 2
15. 15.Which of these numbers is divisible by 9? (a) 2949 (b) 2934 (c) 2937 (d) 2933
16. 16.What is the smallest number you must add to 253 to make it divisible by 4? (a) 3 (b) 2 (c) 1
17. 17.What is the smallest number you must add to 445 to make it divisible by 7? (a) 4 (b) 3 (c) 2 (d) 5
18. 18.Which of these numbers is divisible by 3? (a) 1436 (b) 1432 (c) 1431 (d) 1430
19. 19.Which of these numbers is divisible by 9? (a) 2879 (b) 2862 (c) 2855 (d) 2865
20. 20.Which of these numbers is divisible by 4? (a) 2769 (b) 2768 (c) 2766 (d) 2774
21. 21.What is the units digit of 2^23? (a) 4 (b) 2 (c) 8 (d) 6
22. 22.Which of these numbers is divisible by 3? (a) 2564 (b) 2567 (c) 2570 (d) 2565
23. 23.What is the units digit of 7^40? (a) 9 (b) 7 (c) 1 (d) 3
24. 24.What is the smallest number you must add to 372 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
25. 25.Which of these numbers is divisible by 4? (a) 2750 (b) 2759 (c) 2753 (d) 2752
26. 26.What is the units digit of 7^27? (a) 7 (b) 1 (c) 3 (d) 9
27. 27.Which of these numbers is divisible by 8? (a) 2399 (b) 2384 (c) 2387 (d) 2379
28. 28.What is the smallest number you must add to 242 to make it divisible by 9? (a) 2 (b) 3 (c) 8 (d) 1
29. 29.What is the smallest number you must add to 341 to make it divisible by 10? (a) 1 (b) 9 (c) 8 (d) 7
30. 30.Which of these numbers is divisible by 8? (a) 1607 (b) 1608 (c) 1618 (d) 1610
31. 31.Which of these numbers is divisible by 8? (a) 1519 (b) 1506 (c) 1527 (d) 1512
32. 32.What is the smallest number you must add to 170 to make it divisible by 11? (a) 7 (b) 8 (c) 5 (d) 6
33. 33.Which of these numbers is divisible by 4? (a) 1489 (b) 1488 (c) 1485 (d) 1494
34. 34.What is the smallest number you must add to 199 to make it divisible by 11? (a) 8 (b) 10 (c) 1 (d) 9
35. 35.What is the remainder when 551 is divided by 10? (a) 0 (b) 1 (c) 9 (d) 2
36. 36.What is the units digit of 4^15? (a) 4 (b) 5 (c) 3 (d) 6
37. 37.Which of these numbers is divisible by 3? (a) 2170 (b) 2177 (c) 2172 (d) 2174
38. 38.What is the smallest number you must add to 231 to make it divisible by 10? (a) 8 (b) 9 (c) 7 (d) 1
39. 39.What is the units digit of 9^10? (a) 2 (b) 1 (c) 9 (d) 0
40. 40.What is the units digit of 8^17? (a) 8 (b) 2 (c) 4 (d) 6

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