Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 337 · 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 7^38? (a) 9 (b) 3 (c) 7 (d) 1
2. 2.What is the remainder when 312 is divided by 11? (a) 3 (b) 5 (c) 7 (d) 4
3. 3.What is the units digit of 3^48? (a) 1 (b) 7 (c) 9 (d) 3
4. 4.What is the units digit of 4^59? (a) 4 (b) 3 (c) 5 (d) 6
5. 5.What is the smallest number you must add to 152 to make it divisible by 3? (a) 2 (b) 1
6. 6.What is the units digit of 9^19? (a) 1 (b) 8 (c) 9 (d) 7
7. 7.What is the units digit of 9^26? (a) 0 (b) 1 (c) 9 (d) 2
8. 8.What is the units digit of 3^13? (a) 1 (b) 9 (c) 3 (d) 7
9. 9.What is the smallest number you must add to 635 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 5
10. 10.Which of these numbers is divisible by 6? (a) 915 (b) 919 (c) 911 (d) 912
11. 11.What is the units digit of 7^45? (a) 3 (b) 7 (c) 1 (d) 9
12. 12.What is the smallest number you must add to 349 to make it divisible by 8? (a) 3 (b) 4 (c) 2 (d) 5
13. 13.What is the smallest number you must add to 203 to make it divisible by 4? (a) 2 (b) 3 (c) 1
14. 14.Which of these numbers is divisible by 3? (a) 148 (b) 145 (c) 147 (d) 152
15. 15.Which of these numbers is divisible by 6? (a) 4739 (b) 4734 (c) 4741 (d) 4732
16. 16.What is the smallest number you must add to 366 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
17. 17.What is the units digit of 4^42? (a) 4 (b) 6 (c) 7 (d) 5
18. 18.What is the remainder when 883 is divided by 12? (a) 8 (b) 7 (c) 5 (d) 6
19. 19.What is the smallest number you must add to 192 to make it divisible by 9? (a) 3 (b) 7 (c) 5 (d) 6
20. 20.What is the smallest number you must add to 229 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 4
21. 21.Which of these numbers is divisible by 8? (a) 1466 (b) 1475 (c) 1487 (d) 1472
22. 22.What is the remainder when 928 is divided by 12? (a) 5 (b) 3 (c) 8 (d) 4
23. 23.What is the remainder when 237 is divided by 11? (a) 5 (b) 6 (c) 8 (d) 7
24. 24.What is the units digit of 2^21? (a) 2 (b) 6 (c) 4 (d) 8
25. 25.Which of these numbers is divisible by 6? (a) 1696 (b) 1702 (c) 1691 (d) 1692
26. 26.What is the smallest number you must add to 381 to make it divisible by 12? (a) 3 (b) 4 (c) 2 (d) 9
27. 27.What is the smallest number you must add to 219 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
28. 28.What is the remainder when 255 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
29. 29.What is the units digit of 8^16? (a) 8 (b) 4 (c) 2 (d) 6
30. 30.What is the remainder when 118 is divided by 9? (a) 8 (b) 1 (c) 0 (d) 2
31. 31.What is the remainder when 417 is divided by 6? (a) 5 (b) 2 (c) 3 (d) 4
32. 32.What is the units digit of 9^57? (a) 1 (b) 7 (c) 9 (d) 8
33. 33.Which of these numbers is divisible by 9? (a) 4181 (b) 4175 (c) 4176 (d) 4191
34. 34.What is the smallest number you must add to 118 to make it divisible by 7? (a) 2 (b) 1 (c) 6 (d) 3
35. 35.What is the remainder when 319 is divided by 12? (a) 6 (b) 7 (c) 8 (d) 5
36. 36.What is the units digit of 8^11? (a) 8 (b) 4 (c) 6 (d) 2
37. 37.What is the units digit of 4^46? (a) 6 (b) 5 (c) 4 (d) 7
38. 38.What is the smallest number you must add to 113 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
39. 39.What is the remainder when 61 is divided by 3? (a) 2 (b) 1 (c) 0
40. 40.What is the smallest number you must add to 285 to make it divisible by 11? (a) 2 (b) 1 (c) 10 (d) 3

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