Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 955 · 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 346 to make it divisible by 10? (a) 6 (b) 4 (c) 5 (d) 3
2. 2.What is the remainder when 57 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
3. 3.What is the units digit of 3^39? (a) 7 (b) 3 (c) 9 (d) 1
4. 4.What is the remainder when 86 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
5. 5.Which of these numbers is divisible by 3? (a) 2321 (b) 2316 (c) 2318 (d) 2315
6. 6.What is the units digit of 3^44? (a) 9 (b) 3 (c) 7 (d) 1
7. 7.What is the units digit of 7^28? (a) 9 (b) 3 (c) 1 (d) 7
8. 8.Which of these numbers is divisible by 6? (a) 2003 (b) 2005 (c) 2004 (d) 2014
9. 9.What is the units digit of 9^24? (a) 9 (b) 1 (c) 2 (d) 0
10. 10.Which of these numbers is divisible by 3? (a) 394 (b) 398 (c) 396 (d) 401
11. 11.What is the remainder when 283 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
12. 12.What is the remainder when 301 is divided by 6? (a) 1 (b) 0 (c) 2 (d) 5
13. 13.What is the smallest number you must add to 102 to make it divisible by 4? (a) 3 (b) 2 (c) 1
14. 14.What is the units digit of 7^14? (a) 7 (b) 3 (c) 1 (d) 9
15. 15.What is the smallest number you must add to 471 to make it divisible by 8? (a) 7 (b) 1 (c) 3 (d) 2
16. 16.What is the remainder when 316 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 7
17. 17.Which of these numbers is divisible by 11? (a) 6463 (b) 6439 (c) 6446 (d) 6448
18. 18.What is the smallest number you must add to 377 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
19. 19.What is the smallest number you must add to 554 to make it divisible by 12? (a) 11 (b) 2 (c) 10 (d) 9
20. 20.What is the units digit of 3^56? (a) 1 (b) 3 (c) 7 (d) 9
21. 21.What is the remainder when 302 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
22. 22.What is the remainder when 284 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
23. 23.What is the remainder when 486 is divided by 11? (a) 3 (b) 1 (c) 9 (d) 2
24. 24.What is the remainder when 111 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
25. 25.What is the units digit of 7^48? (a) 3 (b) 1 (c) 9 (d) 7
26. 26.What is the remainder when 661 is divided by 10? (a) 0 (b) 1 (c) 9 (d) 2
27. 27.Which of these numbers is divisible by 6? (a) 1035 (b) 1043 (c) 1032 (d) 1030
28. 28.What is the remainder when 210 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
29. 29.What is the smallest number you must add to 171 to make it divisible by 4? (a) 1 (b) 2 (c) 3
30. 30.What is the smallest number you must add to 439 to make it divisible by 7? (a) 5 (b) 3 (c) 1 (d) 2
31. 31.What is the units digit of 8^24? (a) 6 (b) 2 (c) 4 (d) 8
32. 32.What is the units digit of 4^23? (a) 6 (b) 3 (c) 5 (d) 4
33. 33.What is the smallest number you must add to 97 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
34. 34.Which of these numbers is divisible by 6? (a) 242 (b) 246 (c) 253 (d) 247
35. 35.What is the remainder when 291 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
36. 36.Which of these numbers is divisible by 8? (a) 6303 (b) 6296 (c) 6295 (d) 6307
37. 37.What is the units digit of 4^60? (a) 4 (b) 7 (c) 6 (d) 5
38. 38.Which of these numbers is divisible by 4? (a) 305 (b) 304 (c) 309 (d) 302
39. 39.What is the remainder when 291 is divided by 10? (a) 0 (b) 2 (c) 1 (d) 9
40. 40.What is the smallest number you must add to 372 to make it divisible by 7? (a) 1 (b) 4 (c) 5 (d) 6

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