Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 349 · 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 3^12? (a) 7 (b) 1 (c) 3 (d) 9
2. 2.What is the remainder when 119 is divided by 10? (a) 8 (b) 1 (c) 7 (d) 9
3. 3.Which of these numbers is divisible by 3? (a) 253 (b) 250 (c) 247 (d) 249
4. 4.What is the units digit of 4^13? (a) 4 (b) 3 (c) 6 (d) 5
5. 5.What is the remainder when 296 is divided by 9? (a) 7 (b) 6 (c) 8 (d) 1
6. 6.What is the remainder when 245 is divided by 9? (a) 1 (b) 7 (c) 3 (d) 2
7. 7.What is the units digit of 8^38? (a) 8 (b) 4 (c) 2 (d) 6
8. 8.What is the smallest number you must add to 166 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 1
9. 9.Which of these numbers is divisible by 6? (a) 2160 (b) 2167 (c) 2164 (d) 2157
10. 10.What is the remainder when 105 is divided by 6? (a) 3 (b) 2 (c) 5 (d) 4
11. 11.What is the units digit of 8^23? (a) 6 (b) 4 (c) 8 (d) 2
12. 12.What is the smallest number you must add to 179 to make it divisible by 12? (a) 3 (b) 11 (c) 2 (d) 1
13. 13.What is the smallest number you must add to 113 to make it divisible by 8? (a) 7 (b) 1 (c) 6 (d) 5
14. 14.Which of these numbers is divisible by 6? (a) 5070 (b) 5077 (c) 5072 (d) 5069
15. 15.What is the smallest number you must add to 65 to make it divisible by 3? (a) 1 (b) 2
16. 16.What is the remainder when 103 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
17. 17.What is the units digit of 7^21? (a) 1 (b) 3 (c) 9 (d) 7
18. 18.Which of these numbers is divisible by 9? (a) 1341 (b) 1342 (c) 1335 (d) 1354
19. 19.What is the remainder when 281 is divided by 9? (a) 3 (b) 7 (c) 2 (d) 1
20. 20.Which of these numbers is divisible by 3? (a) 1915 (b) 1921 (c) 1919 (d) 1917
21. 21.What is the smallest number you must add to 465 to make it divisible by 9? (a) 3 (b) 4 (c) 2 (d) 6
22. 22.Which of these numbers is divisible by 8? (a) 6057 (b) 6067 (c) 6064 (d) 6079
23. 23.Which of these numbers is divisible by 3? (a) 2446 (b) 2450 (c) 2448 (d) 2453
24. 24.Which of these numbers is divisible by 11? (a) 7984 (b) 7995 (c) 8002 (d) 7986
25. 25.What is the smallest number you must add to 224 to make it divisible by 3? (a) 1 (b) 2
26. 26.What is the remainder when 646 is divided by 9? (a) 7 (b) 8 (c) 6 (d) 2
27. 27.Which of these numbers is divisible by 9? (a) 2167 (b) 2160 (c) 2172 (d) 2155
28. 28.What is the smallest number you must add to 187 to make it divisible by 8? (a) 6 (b) 4 (c) 5 (d) 3
29. 29.What is the units digit of 2^13? (a) 6 (b) 4 (c) 8 (d) 2
30. 30.What is the smallest number you must add to 138 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
31. 31.What is the remainder when 721 is divided by 10? (a) 2 (b) 9 (c) 0 (d) 1
32. 32.What is the units digit of 8^20? (a) 4 (b) 8 (c) 6 (d) 2
33. 33.Which of these numbers is divisible by 4? (a) 892 (b) 894 (c) 899 (d) 891
34. 34.What is the smallest number you must add to 164 to make it divisible by 3? (a) 2 (b) 1
35. 35.Which of these numbers is divisible by 4? (a) 951 (b) 953 (c) 952 (d) 957
36. 36.What is the remainder when 662 is divided by 9? (a) 6 (b) 4 (c) 7 (d) 5
37. 37.What is the units digit of 7^52? (a) 1 (b) 9 (c) 3 (d) 7
38. 38.Which of these numbers is divisible by 6? (a) 5018 (b) 5013 (c) 5010 (d) 5005
39. 39.What is the smallest number you must add to 89 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
40. 40.Which of these numbers is divisible by 4? (a) 1772 (b) 1778 (c) 1775 (d) 1769

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