Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 921 · 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 143 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
2. 2.What is the units digit of 7^58? (a) 3 (b) 7 (c) 9 (d) 1
3. 3.What is the remainder when 237 is divided by 7? (a) 1 (b) 5 (c) 6 (d) 4
4. 4.What is the remainder when 641 is divided by 8? (a) 7 (b) 1 (c) 2 (d) 0
5. 5.What is the units digit of 8^24? (a) 2 (b) 6 (c) 4 (d) 8
6. 6.What is the smallest number you must add to 92 to make it divisible by 9? (a) 7 (b) 2 (c) 6 (d) 8
7. 7.Which of these numbers is divisible by 9? (a) 1559 (b) 1567 (c) 1581 (d) 1566
8. 8.What is the remainder when 178 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 2
9. 9.Which of these numbers is divisible by 8? (a) 351 (b) 330 (c) 338 (d) 336
10. 10.Which of these numbers is divisible by 9? (a) 2069 (b) 2056 (c) 2048 (d) 2052
11. 11.Which of these numbers is divisible by 8? (a) 4110 (b) 4094 (c) 4096 (d) 4097
12. 12.What is the smallest number you must add to 95 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the remainder when 364 is divided by 10? (a) 6 (b) 5 (c) 3 (d) 4
14. 14.What is the remainder when 295 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
15. 15.Which of these numbers is divisible by 4? (a) 1333 (b) 1325 (c) 1331 (d) 1328
16. 16.What is the smallest number you must add to 196 to make it divisible by 12? (a) 8 (b) 9 (c) 4 (d) 7
17. 17.What is the units digit of 8^40? (a) 6 (b) 2 (c) 4 (d) 8
18. 18.What is the remainder when 452 is divided by 11? (a) 1 (b) 10 (c) 2 (d) 0
19. 19.What is the smallest number you must add to 325 to make it divisible by 7? (a) 5 (b) 4 (c) 3 (d) 6
20. 20.What is the remainder when 516 is divided by 11? (a) 8 (b) 1 (c) 10 (d) 9
21. 21.What is the remainder when 165 is divided by 7? (a) 5 (b) 4 (c) 3 (d) 6
22. 22.What is the remainder when 43 is divided by 3? (a) 0 (b) 1 (c) 2
23. 23.What is the units digit of 2^26? (a) 6 (b) 4 (c) 2 (d) 8
24. 24.Which of these numbers is divisible by 4? (a) 1371 (b) 1368 (c) 1366 (d) 1373
25. 25.What is the units digit of 3^21? (a) 9 (b) 1 (c) 3 (d) 7
26. 26.What is the smallest number you must add to 368 to make it divisible by 10? (a) 1 (b) 8 (c) 3 (d) 2
27. 27.What is the smallest number you must add to 85 to make it divisible by 3? (a) 2 (b) 1
28. 28.What is the units digit of 3^46? (a) 1 (b) 7 (c) 9 (d) 3
29. 29.What is the smallest number you must add to 727 to make it divisible by 11? (a) 8 (b) 1 (c) 9 (d) 10
30. 30.Which of these numbers is divisible by 9? (a) 7861 (b) 7867 (c) 7857 (d) 7851
31. 31.Which of these numbers is divisible by 4? (a) 2794 (b) 2799 (c) 2796 (d) 2803
32. 32.What is the units digit of 3^26? (a) 3 (b) 9 (c) 7 (d) 1
33. 33.What is the units digit of 3^39? (a) 9 (b) 3 (c) 1 (d) 7
34. 34.What is the remainder when 291 is divided by 12? (a) 9 (b) 2 (c) 3 (d) 4
35. 35.What is the smallest number you must add to 716 to make it divisible by 10? (a) 3 (b) 6 (c) 4 (d) 5
36. 36.Which of these numbers is divisible by 9? (a) 7530 (b) 7519 (c) 7515 (d) 7508
37. 37.What is the smallest number you must add to 171 to make it divisible by 7? (a) 5 (b) 4 (c) 3 (d) 6
38. 38.What is the remainder when 435 is divided by 11? (a) 5 (b) 6 (c) 7 (d) 8
39. 39.What is the smallest number you must add to 700 to make it divisible by 9? (a) 2 (b) 1 (c) 3 (d) 7
40. 40.Which of these numbers is divisible by 4? (a) 1568 (b) 1575 (c) 1566 (d) 1569

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