Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 327 · 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 units digit of 8^59? (a) 2 (b) 8 (c) 4 (d) 6
2. 2.What is the smallest number you must add to 419 to make it divisible by 10? (a) 9 (b) 2 (c) 1 (d) 3
3. 3.What is the units digit of 3^48? (a) 3 (b) 9 (c) 7 (d) 1
4. 4.Which of these numbers is divisible by 6? (a) 2038 (b) 2026 (c) 2032 (d) 2028
5. 5.What is the smallest number you must add to 729 to make it divisible by 12? (a) 3 (b) 9 (c) 2 (d) 4
6. 6.What is the smallest number you must add to 203 to make it divisible by 4? (a) 3 (b) 1 (c) 2
7. 7.What is the smallest number you must add to 604 to make it divisible by 10? (a) 7 (b) 5 (c) 6 (d) 4
8. 8.What is the units digit of 3^17? (a) 9 (b) 3 (c) 7 (d) 1
9. 9.Which of these numbers is divisible by 8? (a) 6731 (b) 6738 (c) 6747 (d) 6736
10. 10.What is the units digit of 3^45? (a) 9 (b) 1 (c) 3 (d) 7
11. 11.Which of these numbers is divisible by 6? (a) 2254 (b) 2267 (c) 2256 (d) 2261
12. 12.What is the units digit of 7^53? (a) 9 (b) 1 (c) 7 (d) 3
13. 13.Which of these numbers is divisible by 6? (a) 1372 (b) 1375 (c) 1374 (d) 1384
14. 14.What is the smallest number you must add to 104 to make it divisible by 7? (a) 6 (b) 2 (c) 1 (d) 3
15. 15.What is the smallest number you must add to 301 to make it divisible by 4? (a) 3 (b) 2 (c) 1
16. 16.What is the remainder when 154 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
17. 17.What is the smallest number you must add to 253 to make it divisible by 4? (a) 3 (b) 2 (c) 1
18. 18.What is the smallest number you must add to 166 to make it divisible by 3? (a) 2 (b) 1
19. 19.Which of these numbers is divisible by 3? (a) 562 (b) 559 (c) 558 (d) 556
20. 20.What is the remainder when 116 is divided by 8? (a) 4 (b) 6 (c) 5 (d) 3
21. 21.What is the remainder when 594 is divided by 8? (a) 2 (b) 6 (c) 1 (d) 3
22. 22.Which of these numbers is divisible by 9? (a) 1869 (b) 1884 (c) 1872 (d) 1879
23. 23.What is the units digit of 8^22? (a) 6 (b) 8 (c) 2 (d) 4
24. 24.What is the remainder when 541 is divided by 7? (a) 5 (b) 3 (c) 1 (d) 2
25. 25.What is the units digit of 4^13? (a) 6 (b) 5 (c) 4 (d) 3
26. 26.What is the smallest number you must add to 311 to make it divisible by 4? (a) 2 (b) 3 (c) 1
27. 27.What is the units digit of 2^10? (a) 6 (b) 8 (c) 2 (d) 4
28. 28.What is the remainder when 227 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
29. 29.What is the units digit of 7^51? (a) 1 (b) 3 (c) 9 (d) 7
30. 30.What is the units digit of 2^24? (a) 8 (b) 2 (c) 6 (d) 4
31. 31.What is the units digit of 7^16? (a) 1 (b) 7 (c) 3 (d) 9
32. 32.What is the units digit of 2^56? (a) 6 (b) 4 (c) 8 (d) 2
33. 33.What is the units digit of 9^48? (a) 0 (b) 1 (c) 9 (d) 2
34. 34.What is the smallest number you must add to 295 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 9
35. 35.What is the remainder when 485 is divided by 11? (a) 1 (b) 0 (c) 10 (d) 2
36. 36.What is the units digit of 9^16? (a) 9 (b) 1 (c) 2 (d) 0
37. 37.Which of these numbers is divisible by 11? (a) 9105 (b) 9122 (c) 9112 (d) 9108
38. 38.What is the units digit of 3^12? (a) 3 (b) 1 (c) 7 (d) 9
39. 39.Which of these numbers is divisible by 4? (a) 1936 (b) 1943 (c) 1933 (d) 1939
40. 40.Which of these numbers is divisible by 3? (a) 2078 (b) 2081 (c) 2084 (d) 2079

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