Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 127 · 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 4? (a) 2425 (b) 2421 (c) 2429 (d) 2424
2. 2.What is the units digit of 7^11? (a) 9 (b) 1 (c) 7 (d) 3
3. 3.What is the smallest number you must add to 494 to make it divisible by 10? (a) 6 (b) 4 (c) 7 (d) 5
4. 4.What is the remainder when 203 is divided by 3? (a) 2 (b) 0 (c) 1
5. 5.What is the units digit of 7^10? (a) 7 (b) 1 (c) 3 (d) 9
6. 6.What is the remainder when 417 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
7. 7.Which of these numbers is divisible by 6? (a) 709 (b) 714 (c) 722 (d) 717
8. 8.What is the smallest number you must add to 476 to make it divisible by 10? (a) 6 (b) 5 (c) 3 (d) 4
9. 9.What is the smallest number you must add to 461 to make it divisible by 10? (a) 8 (b) 7 (c) 1 (d) 9
10. 10.Which of these numbers is divisible by 11? (a) 2946 (b) 2955 (c) 2937 (d) 2935
11. 11.Which of these numbers is divisible by 11? (a) 2738 (b) 2752 (c) 2739 (d) 2748
12. 12.What is the remainder when 319 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 3
13. 13.What is the remainder when 74 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
14. 14.What is the remainder when 290 is divided by 7? (a) 5 (b) 4 (c) 3 (d) 2
15. 15.What is the smallest number you must add to 564 to make it divisible by 7? (a) 2 (b) 3 (c) 5 (d) 4
16. 16.Which of these numbers is divisible by 9? (a) 5591 (b) 5573 (c) 5586 (d) 5580
17. 17.Which of these numbers is divisible by 9? (a) 3654 (b) 3646 (c) 3656 (d) 3667
18. 18.What is the units digit of 4^26? (a) 5 (b) 6 (c) 4 (d) 7
19. 19.What is the units digit of 4^58? (a) 5 (b) 4 (c) 6 (d) 7
20. 20.What is the units digit of 3^53? (a) 7 (b) 9 (c) 1 (d) 3
21. 21.What is the remainder when 391 is divided by 12? (a) 5 (b) 6 (c) 7 (d) 8
22. 22.Which of these numbers is divisible by 6? (a) 4397 (b) 4392 (c) 4387 (d) 4399
23. 23.What is the units digit of 3^37? (a) 9 (b) 1 (c) 7 (d) 3
24. 24.What is the units digit of 9^54? (a) 2 (b) 9 (c) 0 (d) 1
25. 25.Which of these numbers is divisible by 6? (a) 362 (b) 355 (c) 370 (d) 360
26. 26.What is the smallest number you must add to 105 to make it divisible by 4? (a) 2 (b) 1 (c) 3
27. 27.Which of these numbers is divisible by 11? (a) 8479 (b) 8457 (c) 8459 (d) 8468
28. 28.What is the units digit of 2^14? (a) 6 (b) 4 (c) 8 (d) 2
29. 29.Which of these numbers is divisible by 8? (a) 1239 (b) 1226 (c) 1224 (d) 1219
30. 30.What is the smallest number you must add to 122 to make it divisible by 9? (a) 5 (b) 3 (c) 4 (d) 6
31. 31.What is the remainder when 49 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
32. 32.Which of these numbers is divisible by 9? (a) 5674 (b) 5687 (c) 5695 (d) 5679
33. 33.What is the remainder when 223 is divided by 3? (a) 0 (b) 2 (c) 1
34. 34.Which of these numbers is divisible by 3? (a) 1280 (b) 1281 (c) 1282 (d) 1286
35. 35.What is the smallest number you must add to 298 to make it divisible by 4? (a) 3 (b) 2 (c) 1
36. 36.What is the units digit of 4^14? (a) 7 (b) 6 (c) 5 (d) 4
37. 37.What is the units digit of 3^11? (a) 3 (b) 7 (c) 9 (d) 1
38. 38.What is the smallest number you must add to 211 to make it divisible by 3? (a) 2 (b) 1
39. 39.Which of these numbers is divisible by 9? (a) 6910 (b) 6902 (c) 6920 (d) 6903
40. 40.What is the remainder when 554 is divided by 8? (a) 2 (b) 6 (c) 3 (d) 1

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