Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 238 · 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.Which of these numbers is divisible by 11? (a) 6527 (b) 6523 (c) 6522 (d) 6542
2. 2.What is the units digit of 3^40? (a) 7 (b) 3 (c) 9 (d) 1
3. 3.What is the units digit of 9^30? (a) 0 (b) 2 (c) 1 (d) 9
4. 4.What is the remainder when 131 is divided by 3? (a) 1 (b) 0 (c) 2
5. 5.What is the smallest number you must add to 41 to make it divisible by 3? (a) 2 (b) 1
6. 6.Which of these numbers is divisible by 6? (a) 947 (b) 941 (c) 932 (d) 936
7. 7.What is the remainder when 113 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
8. 8.What is the units digit of 2^58? (a) 8 (b) 6 (c) 2 (d) 4
9. 9.What is the units digit of 8^24? (a) 2 (b) 6 (c) 4 (d) 8
10. 10.What is the remainder when 303 is divided by 12? (a) 2 (b) 3 (c) 4 (d) 9
11. 11.What is the remainder when 117 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
12. 12.Which of these numbers is divisible by 4? (a) 2244 (b) 2251 (c) 2245 (d) 2243
13. 13.What is the smallest number you must add to 254 to make it divisible by 4? (a) 1 (b) 3 (c) 2
14. 14.Which of these numbers is divisible by 3? (a) 1063 (b) 1066 (c) 1060 (d) 1062
15. 15.What is the smallest number you must add to 308 to make it divisible by 10? (a) 1 (b) 8 (c) 2 (d) 3
16. 16.What is the remainder when 318 is divided by 11? (a) 9 (b) 1 (c) 8 (d) 10
17. 17.What is the units digit of 8^11? (a) 8 (b) 6 (c) 2 (d) 4
18. 18.What is the smallest number you must add to 133 to make it divisible by 3? (a) 1 (b) 2
19. 19.What is the smallest number you must add to 489 to make it divisible by 9? (a) 3 (b) 6 (c) 5 (d) 7
20. 20.Which of these numbers is divisible by 4? (a) 489 (b) 494 (c) 499 (d) 492
21. 21.Which of these numbers is divisible by 11? (a) 6188 (b) 6178 (c) 6194 (d) 6182
22. 22.What is the units digit of 4^19? (a) 5 (b) 6 (c) 3 (d) 4
23. 23.What is the smallest number you must add to 115 to make it divisible by 7? (a) 5 (b) 4 (c) 3 (d) 6
24. 24.Which of these numbers is divisible by 3? (a) 733 (b) 732 (c) 737 (d) 731
25. 25.What is the smallest number you must add to 294 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
26. 26.What is the smallest number you must add to 570 to make it divisible by 9? (a) 5 (b) 6 (c) 7 (d) 3
27. 27.What is the remainder when 491 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 2
28. 28.What is the remainder when 371 is divided by 5? (a) 4 (b) 0 (c) 2 (d) 1
29. 29.What is the units digit of 7^58? (a) 9 (b) 1 (c) 3 (d) 7
30. 30.What is the units digit of 9^40? (a) 2 (b) 9 (c) 0 (d) 1
31. 31.What is the smallest number you must add to 234 to make it divisible by 12? (a) 6 (b) 7 (c) 8 (d) 5
32. 32.What is the units digit of 3^25? (a) 9 (b) 1 (c) 7 (d) 3
33. 33.What is the remainder when 569 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
34. 34.Which of these numbers is divisible by 9? (a) 7710 (b) 7730 (c) 7713 (d) 7721
35. 35.What is the units digit of 7^44? (a) 9 (b) 7 (c) 3 (d) 1
36. 36.What is the smallest number you must add to 325 to make it divisible by 6? (a) 3 (b) 1 (c) 5 (d) 4
37. 37.What is the units digit of 3^58? (a) 3 (b) 1 (c) 7 (d) 9
38. 38.What is the remainder when 97 is divided by 3? (a) 0 (b) 2 (c) 1
39. 39.What is the remainder when 125 is divided by 3? (a) 1 (b) 2 (c) 0
40. 40.What is the smallest number you must add to 74 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2

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