Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 150 · 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.Which of these numbers is divisible by 6? (a) 3681 (b) 3678 (c) 3676 (d) 3686
2. 2.Which of these numbers is divisible by 3? (a) 1468 (b) 1463 (c) 1464 (d) 1465
3. 3.Which of these numbers is divisible by 9? (a) 2468 (b) 2457 (c) 2454 (d) 2460
4. 4.What is the smallest number you must add to 418 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 2
5. 5.What is the units digit of 3^41? (a) 3 (b) 1 (c) 9 (d) 7
6. 6.Which of these numbers is divisible by 3? (a) 1727 (b) 1729 (c) 1725 (d) 1723
7. 7.What is the units digit of 2^42? (a) 6 (b) 2 (c) 4 (d) 8
8. 8.Which of these numbers is divisible by 3? (a) 1015 (b) 1013 (c) 1011 (d) 1010
9. 9.Which of these numbers is divisible by 8? (a) 2976 (b) 2983 (c) 2970 (d) 2989
10. 10.What is the remainder when 183 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
11. 11.What is the smallest number you must add to 137 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
12. 12.Which of these numbers is divisible by 8? (a) 2325 (b) 2316 (c) 2311 (d) 2312
13. 13.What is the remainder when 554 is divided by 11? (a) 3 (b) 5 (c) 4 (d) 7
14. 14.What is the remainder when 110 is divided by 6? (a) 1 (b) 2 (c) 3 (d) 4
15. 15.Which of these numbers is divisible by 4? (a) 878 (b) 885 (c) 881 (d) 880
16. 16.What is the units digit of 7^46? (a) 3 (b) 1 (c) 9 (d) 7
17. 17.What is the remainder when 428 is divided by 10? (a) 7 (b) 9 (c) 2 (d) 8
18. 18.What is the units digit of 3^29? (a) 7 (b) 9 (c) 3 (d) 1
19. 19.What is the remainder when 93 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
20. 20.Which of these numbers is divisible by 6? (a) 4942 (b) 4937 (c) 4928 (d) 4932
21. 21.What is the smallest number you must add to 392 to make it divisible by 10? (a) 2 (b) 8 (c) 7 (d) 9
22. 22.What is the smallest number you must add to 619 to make it divisible by 11? (a) 8 (b) 9 (c) 3 (d) 7
23. 23.What is the smallest number you must add to 167 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
24. 24.What is the remainder when 278 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
25. 25.What is the smallest number you must add to 289 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
26. 26.What is the units digit of 7^60? (a) 1 (b) 9 (c) 7 (d) 3
27. 27.What is the remainder when 309 is divided by 12? (a) 8 (b) 10 (c) 9 (d) 3
28. 28.Which of these numbers is divisible by 3? (a) 132 (b) 134 (c) 130 (d) 136
29. 29.What is the smallest number you must add to 261 to make it divisible by 4? (a) 3 (b) 1 (c) 2
30. 30.What is the remainder when 175 is divided by 3? (a) 0 (b) 1 (c) 2
31. 31.What is the units digit of 9^29? (a) 7 (b) 1 (c) 8 (d) 9
32. 32.What is the smallest number you must add to 407 to make it divisible by 10? (a) 7 (b) 3 (c) 2 (d) 4
33. 33.What is the smallest number you must add to 190 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
34. 34.What is the remainder when 79 is divided by 3? (a) 1 (b) 0 (c) 2
35. 35.What is the units digit of 2^40? (a) 2 (b) 8 (c) 4 (d) 6
36. 36.What is the smallest number you must add to 79 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
37. 37.What is the remainder when 646 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 7
38. 38.Which of these numbers is divisible by 8? (a) 1643 (b) 1653 (c) 1640 (d) 1639
39. 39.Which of these numbers is divisible by 8? (a) 5407 (b) 5392 (c) 5385 (d) 5399
40. 40.What is the units digit of 3^25? (a) 9 (b) 1 (c) 3 (d) 7

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