Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 383 · 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 118 to make it divisible by 4? (a) 1 (b) 2 (c) 3
2. 2.Which of these numbers is divisible by 6? (a) 2940 (b) 2938 (c) 2944 (d) 2947
3. 3.What is the remainder when 436 is divided by 7? (a) 5 (b) 2 (c) 1 (d) 3
4. 4.What is the remainder when 121 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
5. 5.Which of these numbers is divisible by 6? (a) 1540 (b) 1542 (c) 1545 (d) 1549
6. 6.Which of these numbers is divisible by 4? (a) 2154 (b) 2159 (c) 2161 (d) 2156
7. 7.What is the units digit of 9^57? (a) 8 (b) 1 (c) 9 (d) 7
8. 8.Which of these numbers is divisible by 8? (a) 616 (b) 612 (c) 620 (d) 629
9. 9.What is the smallest number you must add to 591 to make it divisible by 9? (a) 4 (b) 2 (c) 6 (d) 3
10. 10.Which of these numbers is divisible by 3? (a) 1131 (b) 1135 (c) 1130 (d) 1133
11. 11.What is the remainder when 87 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
12. 12.What is the units digit of 2^33? (a) 6 (b) 2 (c) 8 (d) 4
13. 13.What is the units digit of 7^54? (a) 1 (b) 3 (c) 7 (d) 9
14. 14.Which of these numbers is divisible by 9? (a) 3487 (b) 3474 (c) 3469 (d) 3476
15. 15.Which of these numbers is divisible by 3? (a) 153 (b) 155 (c) 151 (d) 158
16. 16.Which of these numbers is divisible by 6? (a) 1003 (b) 998 (c) 1009 (d) 1002
17. 17.What is the units digit of 7^16? (a) 7 (b) 1 (c) 3 (d) 9
18. 18.Which of these numbers is divisible by 4? (a) 901 (b) 904 (c) 910 (d) 905
19. 19.Which of these numbers is divisible by 6? (a) 3146 (b) 3142 (c) 3138 (d) 3134
20. 20.What is the smallest number you must add to 883 to make it divisible by 11? (a) 9 (b) 8 (c) 3 (d) 7
21. 21.What is the units digit of 4^10? (a) 4 (b) 6 (c) 7 (d) 5
22. 22.What is the smallest number you must add to 239 to make it divisible by 3? (a) 1 (b) 2
23. 23.What is the remainder when 543 is divided by 8? (a) 7 (b) 6 (c) 1 (d) 5
24. 24.Which of these numbers is divisible by 9? (a) 1960 (b) 1941 (c) 1945 (d) 1944
25. 25.What is the units digit of 7^53? (a) 7 (b) 3 (c) 9 (d) 1
26. 26.What is the smallest number you must add to 103 to make it divisible by 4? (a) 2 (b) 1 (c) 3
27. 27.What is the remainder when 589 is divided by 11? (a) 7 (b) 6 (c) 8 (d) 5
28. 28.What is the remainder when 157 is divided by 3? (a) 1 (b) 0 (c) 2
29. 29.What is the remainder when 399 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
30. 30.What is the smallest number you must add to 225 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 1
31. 31.What is the units digit of 8^54? (a) 4 (b) 6 (c) 8 (d) 2
32. 32.Which of these numbers is divisible by 8? (a) 2440 (b) 2453 (c) 2441 (d) 2436
33. 33.Which of these numbers is divisible by 9? (a) 3015 (b) 3032 (c) 3019 (d) 3013
34. 34.What is the remainder when 357 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
35. 35.What is the smallest number you must add to 305 to make it divisible by 8? (a) 1 (b) 5 (c) 6 (d) 7
36. 36.What is the smallest number you must add to 527 to make it divisible by 10? (a) 4 (b) 7 (c) 2 (d) 3
37. 37.Which of these numbers is divisible by 4? (a) 400 (b) 407 (c) 399 (d) 401
38. 38.Which of these numbers is divisible by 3? (a) 478 (b) 480 (c) 485 (d) 482
39. 39.What is the remainder when 457 is divided by 9? (a) 7 (b) 8 (c) 2 (d) 6
40. 40.What is the smallest number you must add to 114 to make it divisible by 11? (a) 7 (b) 6 (c) 8 (d) 4

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