Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 332 · 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) 1908 (b) 1910 (c) 1905 (d) 1917
2. 2.What is the smallest number you must add to 106 to make it divisible by 3? (a) 1 (b) 2
3. 3.Which of these numbers is divisible by 11? (a) 3305 (b) 3327 (c) 3311 (d) 3317
4. 4.What is the smallest number you must add to 303 to make it divisible by 9? (a) 3 (b) 2 (c) 6 (d) 4
5. 5.What is the remainder when 169 is divided by 8? (a) 7 (b) 0 (c) 2 (d) 1
6. 6.What is the units digit of 2^18? (a) 4 (b) 6 (c) 8 (d) 2
7. 7.What is the units digit of 9^49? (a) 8 (b) 1 (c) 9 (d) 7
8. 8.What is the smallest number you must add to 284 to make it divisible by 12? (a) 5 (b) 8 (c) 3 (d) 4
9. 9.What is the remainder when 471 is divided by 10? (a) 1 (b) 2 (c) 0 (d) 9
10. 10.What is the remainder when 463 is divided by 9? (a) 4 (b) 6 (c) 5 (d) 3
11. 11.What is the units digit of 2^38? (a) 4 (b) 6 (c) 8 (d) 2
12. 12.What is the remainder when 134 is divided by 3? (a) 0 (b) 1 (c) 2
13. 13.What is the units digit of 3^39? (a) 7 (b) 9 (c) 1 (d) 3
14. 14.What is the units digit of 7^46? (a) 9 (b) 7 (c) 3 (d) 1
15. 15.Which of these numbers is divisible by 6? (a) 440 (b) 445 (c) 438 (d) 437
16. 16.What is the units digit of 4^33? (a) 6 (b) 5 (c) 3 (d) 4
17. 17.Which of these numbers is divisible by 4? (a) 302 (b) 295 (c) 296 (d) 297
18. 18.What is the smallest number you must add to 419 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
19. 19.What is the smallest number you must add to 146 to make it divisible by 3? (a) 1 (b) 2
20. 20.What is the remainder when 421 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
21. 21.Which of these numbers is divisible by 3? (a) 398 (b) 393 (c) 395 (d) 391
22. 22.What is the remainder when 328 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
23. 23.What is the smallest number you must add to 251 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 5
24. 24.Which of these numbers is divisible by 11? (a) 4269 (b) 4259 (c) 4257 (d) 4255
25. 25.What is the units digit of 3^25? (a) 3 (b) 9 (c) 7 (d) 1
26. 26.What is the smallest number you must add to 215 to make it divisible by 3? (a) 1 (b) 2
27. 27.Which of these numbers is divisible by 3? (a) 2417 (b) 2410 (c) 2412 (d) 2414
28. 28.What is the units digit of 9^31? (a) 8 (b) 9 (c) 7 (d) 1
29. 29.What is the smallest number you must add to 231 to make it divisible by 8? (a) 1 (b) 7 (c) 2 (d) 3
30. 30.What is the smallest number you must add to 327 to make it divisible by 9? (a) 5 (b) 3 (c) 6 (d) 7
31. 31.What is the smallest number you must add to 455 to make it divisible by 6? (a) 5 (b) 1 (c) 2 (d) 3
32. 32.Which of these numbers is divisible by 8? (a) 6954 (b) 6951 (c) 6943 (d) 6944
33. 33.What is the remainder when 763 is divided by 12? (a) 7 (b) 8 (c) 5 (d) 6
34. 34.Which of these numbers is divisible by 6? (a) 4603 (b) 4611 (c) 4599 (d) 4602
35. 35.What is the units digit of 7^30? (a) 1 (b) 9 (c) 7 (d) 3
36. 36.What is the units digit of 7^58? (a) 9 (b) 3 (c) 1 (d) 7
37. 37.What is the smallest number you must add to 116 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 6
38. 38.What is the units digit of 7^22? (a) 9 (b) 7 (c) 3 (d) 1
39. 39.What is the units digit of 7^34? (a) 9 (b) 1 (c) 3 (d) 7
40. 40.What is the units digit of 3^16? (a) 1 (b) 7 (c) 9 (d) 3

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