Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 32 · 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) 3538 (b) 3540 (c) 3548 (d) 3544
2. 2.Which of these numbers is divisible by 4? (a) 2603 (b) 2605 (c) 2610 (d) 2604
3. 3.Which of these numbers is divisible by 11? (a) 7810 (b) 7826 (c) 7813 (d) 7805
4. 4.What is the smallest number you must add to 534 to make it divisible by 10? (a) 4 (b) 7 (c) 6 (d) 5
5. 5.What is the smallest number you must add to 636 to make it divisible by 11? (a) 2 (b) 9 (c) 3 (d) 1
6. 6.What is the units digit of 3^37? (a) 3 (b) 9 (c) 7 (d) 1
7. 7.What is the units digit of 2^23? (a) 8 (b) 6 (c) 2 (d) 4
8. 8.What is the smallest number you must add to 64 to make it divisible by 6? (a) 2 (b) 4 (c) 3 (d) 1
9. 9.Which of these numbers is divisible by 4? (a) 2390 (b) 2384 (c) 2385 (d) 2381
10. 10.What is the remainder when 528 is divided by 10? (a) 8 (b) 9 (c) 7 (d) 2
11. 11.Which of these numbers is divisible by 3? (a) 1953 (b) 1951 (c) 1957 (d) 1954
12. 12.What is the smallest number you must add to 429 to make it divisible by 9? (a) 6 (b) 4 (c) 3 (d) 2
13. 13.What is the units digit of 3^18? (a) 7 (b) 1 (c) 3 (d) 9
14. 14.What is the units digit of 3^36? (a) 7 (b) 1 (c) 9 (d) 3
15. 15.What is the smallest number you must add to 439 to make it divisible by 7? (a) 3 (b) 5 (c) 2 (d) 1
16. 16.What is the smallest number you must add to 722 to make it divisible by 9? (a) 7 (b) 8 (c) 2 (d) 6
17. 17.Which of these numbers is divisible by 11? (a) 8283 (b) 8289 (c) 8275 (d) 8297
18. 18.What is the remainder when 491 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 2
19. 19.What is the remainder when 82 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
20. 20.Which of these numbers is divisible by 3? (a) 2207 (b) 2200 (c) 2202 (d) 2203
21. 21.What is the smallest number you must add to 175 to make it divisible by 3? (a) 1 (b) 2
22. 22.What is the remainder when 394 is divided by 12? (a) 10 (b) 2 (c) 11 (d) 9
23. 23.What is the remainder when 275 is divided by 7? (a) 2 (b) 1 (c) 3 (d) 5
24. 24.What is the units digit of 4^21? (a) 6 (b) 5 (c) 4 (d) 3
25. 25.Which of these numbers is divisible by 6? (a) 3291 (b) 3295 (c) 3294 (d) 3304
26. 26.What is the remainder when 277 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
27. 27.Which of these numbers is divisible by 4? (a) 1493 (b) 1496 (c) 1497 (d) 1502
28. 28.What is the units digit of 3^54? (a) 7 (b) 3 (c) 9 (d) 1
29. 29.What is the remainder when 754 is divided by 10? (a) 6 (b) 4 (c) 3 (d) 5
30. 30.What is the units digit of 2^39? (a) 4 (b) 2 (c) 8 (d) 6
31. 31.Which of these numbers is divisible by 6? (a) 831 (b) 818 (c) 827 (d) 822
32. 32.What is the units digit of 8^28? (a) 2 (b) 6 (c) 8 (d) 4
33. 33.What is the remainder when 196 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
34. 34.What is the remainder when 153 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
35. 35.What is the units digit of 7^43? (a) 3 (b) 7 (c) 1 (d) 9
36. 36.What is the remainder when 321 is divided by 5? (a) 4 (b) 2 (c) 0 (d) 1
37. 37.What is the units digit of 3^29? (a) 9 (b) 3 (c) 7 (d) 1
38. 38.What is the remainder when 203 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 1
39. 39.What is the units digit of 3^14? (a) 7 (b) 1 (c) 3 (d) 9
40. 40.What is the remainder when 109 is divided by 3? (a) 1 (b) 0 (c) 2

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