Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 29 · 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 smallest number you must add to 136 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 1
2. 2.Which of these numbers is divisible by 6? (a) 2426 (b) 2420 (c) 2434 (d) 2424
3. 3.What is the units digit of 3^25? (a) 9 (b) 7 (c) 1 (d) 3
4. 4.What is the smallest number you must add to 568 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 1
5. 5.What is the smallest number you must add to 170 to make it divisible by 9? (a) 3 (b) 1 (c) 8 (d) 2
6. 6.What is the units digit of 3^26? (a) 1 (b) 7 (c) 9 (d) 3
7. 7.Which of these numbers is divisible by 3? (a) 1841 (b) 1838 (c) 1839 (d) 1843
8. 8.What is the units digit of 4^14? (a) 6 (b) 4 (c) 7 (d) 5
9. 9.What is the remainder when 193 is divided by 3? (a) 1 (b) 0 (c) 2
10. 10.What is the smallest number you must add to 778 to make it divisible by 12? (a) 1 (b) 2 (c) 3 (d) 10
11. 11.What is the units digit of 7^12? (a) 1 (b) 7 (c) 9 (d) 3
12. 12.Which of these numbers is divisible by 9? (a) 6999 (b) 7012 (c) 7002 (d) 7003
13. 13.What is the units digit of 3^42? (a) 7 (b) 9 (c) 3 (d) 1
14. 14.What is the smallest number you must add to 199 to make it divisible by 4? (a) 3 (b) 2 (c) 1
15. 15.Which of these numbers is divisible by 8? (a) 864 (b) 862 (c) 866 (d) 876
16. 16.What is the remainder when 661 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
17. 17.Which of these numbers is divisible by 11? (a) 5839 (b) 5858 (c) 5841 (d) 5849
18. 18.What is the smallest number you must add to 98 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
19. 19.Which of these numbers is divisible by 6? (a) 3438 (b) 3436 (c) 3448 (d) 3443
20. 20.Which of these numbers is divisible by 9? (a) 5421 (b) 5429 (c) 5417 (d) 5418
21. 21.Which of these numbers is divisible by 3? (a) 1652 (b) 1649 (c) 1655 (d) 1650
22. 22.What is the remainder when 374 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 2
23. 23.What is the units digit of 3^37? (a) 1 (b) 7 (c) 9 (d) 3
24. 24.What is the smallest number you must add to 614 to make it divisible by 11? (a) 9 (b) 1 (c) 2 (d) 3
25. 25.What is the units digit of 2^47? (a) 8 (b) 6 (c) 4 (d) 2
26. 26.Which of these numbers is divisible by 11? (a) 7487 (b) 7469 (c) 7460 (d) 7476
27. 27.What is the smallest number you must add to 116 to make it divisible by 3? (a) 2 (b) 1
28. 28.What is the smallest number you must add to 663 to make it divisible by 10? (a) 8 (b) 3 (c) 7 (d) 6
29. 29.What is the remainder when 362 is divided by 9? (a) 3 (b) 2 (c) 1 (d) 7
30. 30.Which of these numbers is divisible by 4? (a) 1189 (b) 1199 (c) 1194 (d) 1192
31. 31.What is the remainder when 184 is divided by 3? (a) 0 (b) 2 (c) 1
32. 32.What is the remainder when 86 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
33. 33.Which of these numbers is divisible by 3? (a) 2528 (b) 2530 (c) 2524 (d) 2526
34. 34.What is the remainder when 712 is divided by 12? (a) 3 (b) 4 (c) 8 (d) 5
35. 35.What is the units digit of 2^26? (a) 2 (b) 6 (c) 4 (d) 8
36. 36.What is the smallest number you must add to 281 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 5
37. 37.What is the smallest number you must add to 274 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
38. 38.What is the remainder when 269 is divided by 8? (a) 5 (b) 3 (c) 6 (d) 4
39. 39.What is the units digit of 2^45? (a) 8 (b) 4 (c) 6 (d) 2
40. 40.What is the smallest number you must add to 688 to make it divisible by 11? (a) 4 (b) 5 (c) 6 (d) 7

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