Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 942 · 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 3^17? (a) 1 (b) 7 (c) 9 (d) 3
2. 2.What is the smallest number you must add to 162 to make it divisible by 7? (a) 6 (b) 5 (c) 1 (d) 4
3. 3.What is the units digit of 2^30? (a) 8 (b) 4 (c) 2 (d) 6
4. 4.Which of these numbers is divisible by 11? (a) 9405 (b) 9401 (c) 9408 (d) 9421
5. 5.What is the smallest number you must add to 259 to make it divisible by 12? (a) 5 (b) 4 (c) 6 (d) 7
6. 6.What is the units digit of 7^15? (a) 7 (b) 9 (c) 3 (d) 1
7. 7.What is the units digit of 7^11? (a) 9 (b) 3 (c) 1 (d) 7
8. 8.Which of these numbers is divisible by 11? (a) 1732 (b) 1742 (c) 1718 (d) 1727
9. 9.What is the smallest number you must add to 382 to make it divisible by 9? (a) 4 (b) 6 (c) 7 (d) 5
10. 10.Which of these numbers is divisible by 4? (a) 2210 (b) 2201 (c) 2207 (d) 2204
11. 11.What is the remainder when 855 is divided by 11? (a) 7 (b) 9 (c) 3 (d) 8
12. 12.Which of these numbers is divisible by 3? (a) 1864 (b) 1859 (c) 1860 (d) 1862
13. 13.What is the smallest number you must add to 298 to make it divisible by 4? (a) 1 (b) 2 (c) 3
14. 14.What is the units digit of 2^20? (a) 2 (b) 6 (c) 4 (d) 8
15. 15.What is the remainder when 242 is divided by 10? (a) 2 (b) 1 (c) 3 (d) 8
16. 16.What is the remainder when 175 is divided by 9? (a) 5 (b) 4 (c) 6 (d) 3
17. 17.What is the smallest number you must add to 556 to make it divisible by 12? (a) 8 (b) 7 (c) 4 (d) 9
18. 18.What is the units digit of 7^22? (a) 7 (b) 1 (c) 9 (d) 3
19. 19.What is the units digit of 8^45? (a) 6 (b) 4 (c) 2 (d) 8
20. 20.What is the smallest number you must add to 229 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 9
21. 21.What is the remainder when 301 is divided by 10? (a) 9 (b) 2 (c) 1 (d) 0
22. 22.Which of these numbers is divisible by 3? (a) 2632 (b) 2630 (c) 2635 (d) 2631
23. 23.What is the remainder when 551 is divided by 7? (a) 2 (b) 5 (c) 4 (d) 6
24. 24.What is the units digit of 9^39? (a) 8 (b) 9 (c) 1 (d) 7
25. 25.Which of these numbers is divisible by 3? (a) 992 (b) 993 (c) 995 (d) 998
26. 26.What is the units digit of 8^17? (a) 4 (b) 6 (c) 8 (d) 2
27. 27.Which of these numbers is divisible by 3? (a) 1563 (b) 1562 (c) 1564 (d) 1568
28. 28.Which of these numbers is divisible by 4? (a) 2420 (b) 2419 (c) 2423 (d) 2425
29. 29.What is the smallest number you must add to 149 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
30. 30.Which of these numbers is divisible by 9? (a) 5057 (b) 5039 (c) 5040 (d) 5042
31. 31.What is the units digit of 8^14? (a) 2 (b) 6 (c) 8 (d) 4
32. 32.What is the units digit of 3^34? (a) 7 (b) 9 (c) 1 (d) 3
33. 33.Which of these numbers is divisible by 9? (a) 7160 (b) 7155 (c) 7168 (d) 7148
34. 34.Which of these numbers is divisible by 11? (a) 1357 (b) 1343 (c) 1337 (d) 1342
35. 35.What is the units digit of 2^18? (a) 4 (b) 6 (c) 8 (d) 2
36. 36.What is the units digit of 7^41? (a) 1 (b) 7 (c) 3 (d) 9
37. 37.Which of these numbers is divisible by 6? (a) 1730 (b) 1722 (c) 1719 (d) 1726
38. 38.What is the smallest number you must add to 137 to make it divisible by 3? (a) 1 (b) 2
39. 39.What is the smallest number you must add to 205 to make it divisible by 4? (a) 3 (b) 2 (c) 1
40. 40.Which of these numbers is divisible by 3? (a) 1098 (b) 1102 (c) 1096 (d) 1100

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