Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 76 · 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 smallest number you must add to 274 to make it divisible by 8? (a) 7 (b) 2 (c) 6 (d) 5
2. 2.What is the units digit of 4^60? (a) 7 (b) 5 (c) 4 (d) 6
3. 3.What is the remainder when 161 is divided by 3? (a) 2 (b) 1 (c) 0
4. 4.What is the smallest number you must add to 147 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
5. 5.What is the units digit of 4^46? (a) 7 (b) 5 (c) 4 (d) 6
6. 6.What is the units digit of 3^16? (a) 1 (b) 7 (c) 3 (d) 9
7. 7.What is the smallest number you must add to 309 to make it divisible by 4? (a) 3 (b) 2 (c) 1
8. 8.What is the smallest number you must add to 134 to make it divisible by 7? (a) 4 (b) 5 (c) 1 (d) 6
9. 9.What is the smallest number you must add to 82 to make it divisible by 7? (a) 1 (b) 2 (c) 3 (d) 5
10. 10.What is the units digit of 3^17? (a) 3 (b) 7 (c) 9 (d) 1
11. 11.What is the units digit of 2^19? (a) 6 (b) 4 (c) 2 (d) 8
12. 12.What is the smallest number you must add to 121 to make it divisible by 3? (a) 1 (b) 2
13. 13.Which of these numbers is divisible by 6? (a) 4216 (b) 4220 (c) 4209 (d) 4212
14. 14.What is the remainder when 431 is divided by 6? (a) 1 (b) 4 (c) 5 (d) 3
15. 15.Which of these numbers is divisible by 4? (a) 2006 (b) 2009 (c) 2004 (d) 2002
16. 16.Which of these numbers is divisible by 4? (a) 2216 (b) 2217 (c) 2221 (d) 2215
17. 17.Which of these numbers is divisible by 8? (a) 6480 (b) 6476 (c) 6495 (d) 6483
18. 18.What is the smallest number you must add to 192 to make it divisible by 9? (a) 6 (b) 7 (c) 5 (d) 3
19. 19.What is the units digit of 7^31? (a) 9 (b) 7 (c) 1 (d) 3
20. 20.What is the smallest number you must add to 158 to make it divisible by 7? (a) 5 (b) 2 (c) 4 (d) 3
21. 21.What is the units digit of 7^46? (a) 1 (b) 7 (c) 9 (d) 3
22. 22.Which of these numbers is divisible by 9? (a) 7239 (b) 7248 (c) 7245 (d) 7261
23. 23.What is the remainder when 317 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
24. 24.What is the smallest number you must add to 155 to make it divisible by 10? (a) 6 (b) 4 (c) 7 (d) 5
25. 25.What is the units digit of 8^51? (a) 6 (b) 2 (c) 8 (d) 4
26. 26.What is the units digit of 3^58? (a) 3 (b) 7 (c) 9 (d) 1
27. 27.What is the units digit of 2^15? (a) 8 (b) 4 (c) 6 (d) 2
28. 28.What is the units digit of 7^19? (a) 1 (b) 9 (c) 3 (d) 7
29. 29.Which of these numbers is divisible by 11? (a) 7275 (b) 7302 (c) 7282 (d) 7292
30. 30.What is the units digit of 7^56? (a) 9 (b) 7 (c) 3 (d) 1
31. 31.What is the units digit of 2^22? (a) 4 (b) 6 (c) 8 (d) 2
32. 32.What is the smallest number you must add to 299 to make it divisible by 7? (a) 1 (b) 3 (c) 5 (d) 2
33. 33.What is the smallest number you must add to 568 to make it divisible by 10? (a) 8 (b) 3 (c) 1 (d) 2
34. 34.What is the remainder when 498 is divided by 7? (a) 1 (b) 2 (c) 6 (d) 0
35. 35.Which of these numbers is divisible by 9? (a) 3628 (b) 3650 (c) 3636 (d) 3643
36. 36.What is the smallest number you must add to 127 to make it divisible by 4? (a) 2 (b) 1 (c) 3
37. 37.What is the units digit of 9^39? (a) 1 (b) 8 (c) 7 (d) 9
38. 38.What is the units digit of 3^23? (a) 1 (b) 9 (c) 7 (d) 3
39. 39.Which of these numbers is divisible by 11? (a) 7898 (b) 7903 (c) 7913 (d) 7897
40. 40.What is the units digit of 8^17? (a) 4 (b) 8 (c) 2 (d) 6

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