Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 839 · 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 7^21? (a) 9 (b) 7 (c) 3 (d) 1
2. 2.What is the smallest number you must add to 386 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 1
3. 3.What is the smallest number you must add to 56 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the remainder when 102 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
5. 5.What is the units digit of 2^25? (a) 8 (b) 2 (c) 4 (d) 6
6. 6.What is the units digit of 3^10? (a) 1 (b) 3 (c) 7 (d) 9
7. 7.Which of these numbers is divisible by 6? (a) 2765 (b) 2760 (c) 2757 (d) 2770
8. 8.What is the units digit of 3^14? (a) 3 (b) 7 (c) 9 (d) 1
9. 9.What is the remainder when 332 is divided by 11? (a) 2 (b) 1 (c) 9 (d) 3
10. 10.What is the smallest number you must add to 196 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
11. 11.Which of these numbers is divisible by 4? (a) 3164 (b) 3161 (c) 3170 (d) 3166
12. 12.What is the units digit of 3^50? (a) 3 (b) 7 (c) 1 (d) 9
13. 13.What is the smallest number you must add to 460 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 5
14. 14.Which of these numbers is divisible by 3? (a) 476 (b) 481 (c) 478 (d) 477
15. 15.Which of these numbers is divisible by 8? (a) 2847 (b) 2826 (c) 2837 (d) 2832
16. 16.Which of these numbers is divisible by 11? (a) 8346 (b) 8329 (c) 8323 (d) 8327
17. 17.What is the units digit of 2^35? (a) 8 (b) 6 (c) 4 (d) 2
18. 18.What is the units digit of 4^39? (a) 6 (b) 3 (c) 4 (d) 5
19. 19.What is the units digit of 2^27? (a) 6 (b) 8 (c) 2 (d) 4
20. 20.What is the smallest number you must add to 173 to make it divisible by 3? (a) 2 (b) 1
21. 21.What is the units digit of 4^31? (a) 3 (b) 4 (c) 6 (d) 5
22. 22.Which of these numbers is divisible by 4? (a) 1822 (b) 1815 (c) 1818 (d) 1816
23. 23.What is the units digit of 8^22? (a) 2 (b) 4 (c) 6 (d) 8
24. 24.What is the remainder when 599 is divided by 8? (a) 1 (b) 6 (c) 5 (d) 7
25. 25.What is the smallest number you must add to 566 to make it divisible by 12? (a) 11 (b) 2 (c) 9 (d) 10
26. 26.What is the units digit of 8^45? (a) 4 (b) 8 (c) 6 (d) 2
27. 27.What is the remainder when 209 is divided by 8? (a) 1 (b) 7 (c) 0 (d) 2
28. 28.What is the units digit of 9^58? (a) 9 (b) 1 (c) 0 (d) 2
29. 29.What is the remainder when 164 is divided by 9? (a) 2 (b) 1 (c) 3 (d) 7
30. 30.Which of these numbers is divisible by 6? (a) 4571 (b) 4564 (c) 4577 (d) 4566
31. 31.Which of these numbers is divisible by 8? (a) 2447 (b) 2452 (c) 2460 (d) 2448
32. 32.What is the units digit of 2^50? (a) 6 (b) 2 (c) 8 (d) 4
33. 33.What is the remainder when 248 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 2
34. 34.Which of these numbers is divisible by 6? (a) 293 (b) 294 (c) 296 (d) 305
35. 35.What is the smallest number you must add to 945 to make it divisible by 12? (a) 3 (b) 9 (c) 2 (d) 4
36. 36.What is the smallest number you must add to 431 to make it divisible by 6? (a) 2 (b) 1 (c) 5 (d) 3
37. 37.What is the smallest number you must add to 427 to make it divisible by 12? (a) 6 (b) 5 (c) 4 (d) 7
38. 38.Which of these numbers is divisible by 6? (a) 240 (b) 235 (c) 242 (d) 250
39. 39.Which of these numbers is divisible by 6? (a) 2669 (b) 2663 (c) 2671 (d) 2664
40. 40.What is the units digit of 2^42? (a) 2 (b) 8 (c) 6 (d) 4

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