Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 961 · 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 remainder when 253 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
2. 2.What is the remainder when 160 is divided by 9? (a) 2 (b) 6 (c) 8 (d) 7
3. 3.Which of these numbers is divisible by 6? (a) 1314 (b) 1315 (c) 1312 (d) 1321
4. 4.What is the remainder when 649 is divided by 12? (a) 11 (b) 2 (c) 0 (d) 1
5. 5.What is the smallest number you must add to 578 to make it divisible by 12? (a) 10 (b) 11 (c) 2 (d) 9
6. 6.What is the smallest number you must add to 402 to make it divisible by 10? (a) 2 (b) 9 (c) 8 (d) 7
7. 7.What is the smallest number you must add to 406 to make it divisible by 10? (a) 3 (b) 4 (c) 5 (d) 6
8. 8.What is the remainder when 349 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
9. 9.What is the smallest number you must add to 307 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
10. 10.What is the smallest number you must add to 47 to make it divisible by 4? (a) 3 (b) 2 (c) 1
11. 11.What is the units digit of 7^34? (a) 7 (b) 9 (c) 1 (d) 3
12. 12.Which of these numbers is divisible by 6? (a) 358 (b) 367 (c) 361 (d) 360
13. 13.What is the smallest number you must add to 149 to make it divisible by 4? (a) 1 (b) 2 (c) 3
14. 14.What is the units digit of 7^21? (a) 1 (b) 7 (c) 9 (d) 3
15. 15.What is the smallest number you must add to 595 to make it divisible by 10? (a) 7 (b) 4 (c) 6 (d) 5
16. 16.What is the units digit of 8^60? (a) 8 (b) 4 (c) 2 (d) 6
17. 17.Which of these numbers is divisible by 11? (a) 4417 (b) 4404 (c) 4411 (d) 4430
18. 18.What is the units digit of 7^51? (a) 1 (b) 9 (c) 3 (d) 7
19. 19.What is the units digit of 4^31? (a) 6 (b) 3 (c) 4 (d) 5
20. 20.Which of these numbers is divisible by 4? (a) 2279 (b) 2283 (c) 2280 (d) 2286
21. 21.What is the remainder when 107 is divided by 6? (a) 3 (b) 1 (c) 4 (d) 5
22. 22.What is the remainder when 111 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
23. 23.What is the units digit of 2^51? (a) 6 (b) 8 (c) 4 (d) 2
24. 24.Which of these numbers is divisible by 6? (a) 1566 (b) 1574 (c) 1570 (d) 1565
25. 25.What is the remainder when 211 is divided by 5? (a) 1 (b) 0 (c) 4 (d) 2
26. 26.What is the smallest number you must add to 460 to make it divisible by 11? (a) 9 (b) 2 (c) 1 (d) 3
27. 27.What is the remainder when 305 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
28. 28.Which of these numbers is divisible by 8? (a) 2041 (b) 2053 (c) 2034 (d) 2040
29. 29.Which of these numbers is divisible by 9? (a) 459 (b) 464 (c) 476 (d) 457
30. 30.What is the smallest number you must add to 382 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 3
31. 31.What is the remainder when 55 is divided by 3? (a) 1 (b) 0 (c) 2
32. 32.What is the units digit of 2^46? (a) 2 (b) 6 (c) 4 (d) 8
33. 33.What is the units digit of 8^40? (a) 8 (b) 6 (c) 4 (d) 2
34. 34.Which of these numbers is divisible by 8? (a) 1557 (b) 1545 (c) 1538 (d) 1544
35. 35.Which of these numbers is divisible by 8? (a) 1294 (b) 1296 (c) 1300 (d) 1311
36. 36.What is the units digit of 7^39? (a) 3 (b) 1 (c) 7 (d) 9
37. 37.What is the remainder when 200 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
38. 38.What is the smallest number you must add to 112 to make it divisible by 10? (a) 7 (b) 8 (c) 2 (d) 9
39. 39.What is the remainder when 307 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1
40. 40.What is the remainder when 315 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0

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