Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 422 · 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 194 to make it divisible by 9? (a) 6 (b) 3 (c) 4 (d) 5
2. 2.What is the remainder when 569 is divided by 11? (a) 7 (b) 8 (c) 3 (d) 9
3. 3.What is the units digit of 7^51? (a) 1 (b) 7 (c) 9 (d) 3
4. 4.What is the remainder when 335 is divided by 9? (a) 2 (b) 7 (c) 1 (d) 3
5. 5.What is the smallest number you must add to 293 to make it divisible by 4? (a) 3 (b) 2 (c) 1
6. 6.What is the smallest number you must add to 201 to make it divisible by 9? (a) 5 (b) 3 (c) 6 (d) 7
7. 7.What is the smallest number you must add to 86 to make it divisible by 4? (a) 3 (b) 2 (c) 1
8. 8.What is the remainder when 139 is divided by 10? (a) 7 (b) 9 (c) 1 (d) 8
9. 9.What is the remainder when 489 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
10. 10.What is the remainder when 542 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 2
11. 11.Which of these numbers is divisible by 11? (a) 2906 (b) 2899 (c) 2904 (d) 2916
12. 12.What is the remainder when 188 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
13. 13.What is the units digit of 7^58? (a) 9 (b) 1 (c) 3 (d) 7
14. 14.What is the remainder when 247 is divided by 8? (a) 5 (b) 1 (c) 7 (d) 6
15. 15.What is the units digit of 2^58? (a) 2 (b) 8 (c) 6 (d) 4
16. 16.Which of these numbers is divisible by 4? (a) 3341 (b) 3345 (c) 3338 (d) 3340
17. 17.What is the remainder when 627 is divided by 12? (a) 4 (b) 3 (c) 2 (d) 9
18. 18.Which of these numbers is divisible by 8? (a) 1402 (b) 1400 (c) 1409 (d) 1396
19. 19.Which of these numbers is divisible by 4? (a) 1310 (b) 1308 (c) 1313 (d) 1305
20. 20.What is the smallest number you must add to 377 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 1
21. 21.What is the smallest number you must add to 217 to make it divisible by 8? (a) 6 (b) 5 (c) 1 (d) 7
22. 22.What is the units digit of 8^46? (a) 8 (b) 4 (c) 6 (d) 2
23. 23.What is the units digit of 7^19? (a) 9 (b) 7 (c) 3 (d) 1
24. 24.What is the smallest number you must add to 235 to make it divisible by 4? (a) 1 (b) 2 (c) 3
25. 25.Which of these numbers is divisible by 11? (a) 9681 (b) 9695 (c) 9709 (d) 9691
26. 26.Which of these numbers is divisible by 11? (a) 5661 (b) 5670 (c) 5654 (d) 5653
27. 27.Which of these numbers is divisible by 4? (a) 2838 (b) 2836 (c) 2842 (d) 2835
28. 28.What is the units digit of 3^60? (a) 7 (b) 1 (c) 3 (d) 9
29. 29.What is the units digit of 9^48? (a) 2 (b) 0 (c) 9 (d) 1
30. 30.What is the units digit of 4^22? (a) 5 (b) 6 (c) 7 (d) 4
31. 31.What is the units digit of 7^37? (a) 9 (b) 3 (c) 1 (d) 7
32. 32.Which of these numbers is divisible by 6? (a) 1375 (b) 1382 (c) 1388 (d) 1380
33. 33.Which of these numbers is divisible by 8? (a) 4468 (b) 4472 (c) 4474 (d) 4482
34. 34.What is the remainder when 133 is divided by 8? (a) 6 (b) 3 (c) 5 (d) 4
35. 35.What is the smallest number you must add to 391 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
36. 36.What is the smallest number you must add to 187 to make it divisible by 8? (a) 4 (b) 3 (c) 6 (d) 5
37. 37.What is the smallest number you must add to 131 to make it divisible by 3? (a) 2 (b) 1
38. 38.What is the smallest number you must add to 86 to make it divisible by 3? (a) 2 (b) 1
39. 39.What is the remainder when 617 is divided by 9? (a) 5 (b) 6 (c) 4 (d) 7
40. 40.What is the units digit of 7^49? (a) 9 (b) 7 (c) 1 (d) 3

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