Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 893 · 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 4^40? (a) 5 (b) 7 (c) 6 (d) 4
2. 2.Which of these numbers is divisible by 4? (a) 3241 (b) 3247 (c) 3239 (d) 3240
3. 3.What is the smallest number you must add to 221 to make it divisible by 11? (a) 9 (b) 1 (c) 10 (d) 8
4. 4.Which of these numbers is divisible by 4? (a) 3488 (b) 3486 (c) 3491 (d) 3494
5. 5.What is the smallest number you must add to 309 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
6. 6.What is the remainder when 402 is divided by 9? (a) 5 (b) 7 (c) 6 (d) 3
7. 7.What is the remainder when 91 is divided by 3? (a) 0 (b) 2 (c) 1
8. 8.What is the units digit of 9^60? (a) 9 (b) 0 (c) 2 (d) 1
9. 9.What is the remainder when 617 is divided by 10? (a) 8 (b) 3 (c) 7 (d) 6
10. 10.Which of these numbers is divisible by 3? (a) 874 (b) 868 (c) 870 (d) 872
11. 11.What is the smallest number you must add to 207 to make it divisible by 7? (a) 5 (b) 4 (c) 2 (d) 3
12. 12.What is the smallest number you must add to 275 to make it divisible by 4? (a) 3 (b) 2 (c) 1
13. 13.What is the units digit of 2^44? (a) 6 (b) 4 (c) 2 (d) 8
14. 14.What is the units digit of 7^47? (a) 1 (b) 9 (c) 3 (d) 7
15. 15.What is the units digit of 3^46? (a) 3 (b) 1 (c) 9 (d) 7
16. 16.Which of these numbers is divisible by 11? (a) 9680 (b) 9681 (c) 9699 (d) 9676
17. 17.What is the smallest number you must add to 422 to make it divisible by 12? (a) 9 (b) 11 (c) 2 (d) 10
18. 18.Which of these numbers is divisible by 9? (a) 732 (b) 724 (c) 744 (d) 729
19. 19.What is the smallest number you must add to 119 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 9
20. 20.What is the smallest number you must add to 456 to make it divisible by 10? (a) 3 (b) 5 (c) 6 (d) 4
21. 21.What is the smallest number you must add to 43 to make it divisible by 3? (a) 2 (b) 1
22. 22.What is the units digit of 4^41? (a) 5 (b) 6 (c) 4 (d) 3
23. 23.What is the smallest number you must add to 262 to make it divisible by 8? (a) 2 (b) 1 (c) 6 (d) 3
24. 24.Which of these numbers is divisible by 9? (a) 2851 (b) 2859 (c) 2863 (d) 2853
25. 25.What is the smallest number you must add to 641 to make it divisible by 10? (a) 1 (b) 9 (c) 8 (d) 7
26. 26.What is the remainder when 377 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
27. 27.Which of these numbers is divisible by 4? (a) 999 (b) 995 (c) 1003 (d) 996
28. 28.What is the smallest number you must add to 470 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 2
29. 29.Which of these numbers is divisible by 8? (a) 6708 (b) 6714 (c) 6724 (d) 6712
30. 30.What is the units digit of 3^50? (a) 9 (b) 1 (c) 3 (d) 7
31. 31.What is the smallest number you must add to 112 to make it divisible by 3? (a) 1 (b) 2
32. 32.What is the remainder when 307 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
33. 33.What is the units digit of 3^30? (a) 9 (b) 3 (c) 7 (d) 1
34. 34.What is the units digit of 9^11? (a) 8 (b) 9 (c) 7 (d) 1
35. 35.What is the smallest number you must add to 458 to make it divisible by 10? (a) 8 (b) 1 (c) 2 (d) 3
36. 36.What is the units digit of 3^24? (a) 7 (b) 9 (c) 3 (d) 1
37. 37.What is the remainder when 251 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
38. 38.What is the units digit of 2^14? (a) 6 (b) 8 (c) 2 (d) 4
39. 39.Which of these numbers is divisible by 8? (a) 1432 (b) 1445 (c) 1438 (d) 1425
40. 40.What is the units digit of 4^24? (a) 4 (b) 6 (c) 5 (d) 7

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