Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 309 · 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 489 to make it divisible by 7? (a) 2 (b) 3 (c) 6 (d) 1
2. 2.Which of these numbers is divisible by 9? (a) 4565 (b) 4563 (c) 4560 (d) 4573
3. 3.What is the units digit of 2^15? (a) 2 (b) 8 (c) 6 (d) 4
4. 4.What is the smallest number you must add to 311 to make it divisible by 4? (a) 3 (b) 1 (c) 2
5. 5.What is the smallest number you must add to 47 to make it divisible by 3? (a) 1 (b) 2
6. 6.What is the units digit of 2^51? (a) 6 (b) 8 (c) 2 (d) 4
7. 7.What is the units digit of 4^50? (a) 5 (b) 7 (c) 4 (d) 6
8. 8.Which of these numbers is divisible by 3? (a) 491 (b) 494 (c) 492 (d) 496
9. 9.What is the units digit of 3^55? (a) 1 (b) 9 (c) 3 (d) 7
10. 10.Which of these numbers is divisible by 6? (a) 2886 (b) 2895 (c) 2882 (d) 2889
11. 11.Which of these numbers is divisible by 4? (a) 2977 (b) 2972 (c) 2975 (d) 2971
12. 12.What is the smallest number you must add to 397 to make it divisible by 9? (a) 6 (b) 8 (c) 7 (d) 1
13. 13.What is the smallest number you must add to 705 to make it divisible by 9? (a) 7 (b) 3 (c) 5 (d) 6
14. 14.What is the smallest number you must add to 248 to make it divisible by 9? (a) 3 (b) 5 (c) 4 (d) 6
15. 15.What is the remainder when 171 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
16. 16.What is the units digit of 9^53? (a) 9 (b) 7 (c) 8 (d) 1
17. 17.What is the units digit of 8^24? (a) 2 (b) 8 (c) 6 (d) 4
18. 18.What is the units digit of 2^54? (a) 8 (b) 6 (c) 4 (d) 2
19. 19.What is the smallest number you must add to 205 to make it divisible by 4? (a) 3 (b) 1 (c) 2
20. 20.Which of these numbers is divisible by 11? (a) 3163 (b) 3152 (c) 3146 (d) 3139
21. 21.What is the smallest number you must add to 118 to make it divisible by 7? (a) 1 (b) 6 (c) 2 (d) 3
22. 22.Which of these numbers is divisible by 3? (a) 1110 (b) 1114 (c) 1112 (d) 1108
23. 23.Which of these numbers is divisible by 4? (a) 1477 (b) 1475 (c) 1482 (d) 1476
24. 24.What is the remainder when 766 is divided by 10? (a) 5 (b) 7 (c) 6 (d) 4
25. 25.What is the units digit of 3^59? (a) 9 (b) 7 (c) 1 (d) 3
26. 26.What is the units digit of 8^27? (a) 2 (b) 4 (c) 6 (d) 8
27. 27.Which of these numbers is divisible by 3? (a) 1447 (b) 1446 (c) 1444 (d) 1450
28. 28.What is the smallest number you must add to 526 to make it divisible by 10? (a) 5 (b) 4 (c) 6 (d) 3
29. 29.What is the units digit of 8^55? (a) 8 (b) 2 (c) 4 (d) 6
30. 30.What is the smallest number you must add to 199 to make it divisible by 9? (a) 7 (b) 1 (c) 8 (d) 6
31. 31.What is the smallest number you must add to 526 to make it divisible by 7? (a) 1 (b) 5 (c) 6 (d) 4
32. 32.Which of these numbers is divisible by 3? (a) 485 (b) 486 (c) 488 (d) 490
33. 33.What is the smallest number you must add to 199 to make it divisible by 3? (a) 1 (b) 2
34. 34.What is the remainder when 175 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
35. 35.What is the smallest number you must add to 294 to make it divisible by 4? (a) 3 (b) 2 (c) 1
36. 36.What is the units digit of 3^51? (a) 9 (b) 3 (c) 1 (d) 7
37. 37.What is the remainder when 163 is divided by 9? (a) 8 (b) 1 (c) 0 (d) 2
38. 38.What is the remainder when 145 is divided by 3? (a) 2 (b) 1 (c) 0
39. 39.What is the remainder when 257 is divided by 11? (a) 5 (b) 4 (c) 7 (d) 3
40. 40.Which of these numbers is divisible by 8? (a) 4959 (b) 4946 (c) 4941 (d) 4944

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