Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 52 · 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 11? (a) 5247 (b) 5240 (c) 5249 (d) 5260
2. 2.Which of these numbers is divisible by 3? (a) 509 (b) 514 (c) 510 (d) 511
3. 3.What is the units digit of 3^35? (a) 1 (b) 7 (c) 3 (d) 9
4. 4.What is the remainder when 403 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
5. 5.What is the smallest number you must add to 211 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 1
6. 6.What is the units digit of 2^16? (a) 8 (b) 2 (c) 4 (d) 6
7. 7.What is the remainder when 121 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
8. 8.What is the smallest number you must add to 41 to make it divisible by 3? (a) 2 (b) 1
9. 9.What is the remainder when 435 is divided by 7? (a) 2 (b) 6 (c) 0 (d) 1
10. 10.What is the remainder when 260 is divided by 7? (a) 6 (b) 1 (c) 2 (d) 0
11. 11.What is the units digit of 4^41? (a) 6 (b) 3 (c) 5 (d) 4
12. 12.What is the units digit of 2^35? (a) 4 (b) 8 (c) 6 (d) 2
13. 13.What is the remainder when 228 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
14. 14.What is the remainder when 166 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 4
15. 15.Which of these numbers is divisible by 4? (a) 2117 (b) 2121 (c) 2116 (d) 2115
16. 16.What is the units digit of 4^57? (a) 4 (b) 5 (c) 3 (d) 6
17. 17.What is the remainder when 196 is divided by 3? (a) 2 (b) 1 (c) 0
18. 18.Which of these numbers is divisible by 6? (a) 4927 (b) 4943 (c) 4934 (d) 4932
19. 19.What is the remainder when 46 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
20. 20.What is the units digit of 8^29? (a) 8 (b) 6 (c) 4 (d) 2
21. 21.What is the units digit of 8^11? (a) 2 (b) 8 (c) 4 (d) 6
22. 22.Which of these numbers is divisible by 3? (a) 1724 (b) 1726 (c) 1721 (d) 1722
23. 23.What is the remainder when 241 is divided by 11? (a) 8 (b) 10 (c) 1 (d) 9
24. 24.Which of these numbers is divisible by 9? (a) 4324 (b) 4309 (c) 4311 (d) 4314
25. 25.Which of these numbers is divisible by 4? (a) 2742 (b) 2744 (c) 2746 (d) 2749
26. 26.Which of these numbers is divisible by 11? (a) 7468 (b) 7487 (c) 7470 (d) 7469
27. 27.Which of these numbers is divisible by 8? (a) 3568 (b) 3562 (c) 3577 (d) 3572
28. 28.Which of these numbers is divisible by 4? (a) 1934 (b) 1932 (c) 1938 (d) 1931
29. 29.What is the units digit of 3^44? (a) 3 (b) 9 (c) 7 (d) 1
30. 30.What is the smallest number you must add to 101 to make it divisible by 4? (a) 3 (b) 1 (c) 2
31. 31.What is the smallest number you must add to 131 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
32. 32.Which of these numbers is divisible by 6? (a) 5359 (b) 5372 (c) 5364 (d) 5365
33. 33.What is the remainder when 128 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 1
34. 34.What is the remainder when 145 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
35. 35.What is the units digit of 3^29? (a) 3 (b) 1 (c) 9 (d) 7
36. 36.What is the units digit of 7^11? (a) 7 (b) 9 (c) 1 (d) 3
37. 37.Which of these numbers is divisible by 8? (a) 1799 (b) 1784 (c) 1787 (d) 1777
38. 38.What is the remainder when 155 is divided by 10? (a) 6 (b) 7 (c) 5 (d) 4
39. 39.What is the units digit of 3^31? (a) 9 (b) 7 (c) 1 (d) 3
40. 40.What is the units digit of 2^34? (a) 4 (b) 6 (c) 8 (d) 2

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