Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 786 · 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 313 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 1
2. 2.What is the units digit of 2^37? (a) 8 (b) 2 (c) 4 (d) 6
3. 3.What is the smallest number you must add to 260 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
4. 4.What is the remainder when 106 is divided by 3? (a) 2 (b) 1 (c) 0
5. 5.What is the units digit of 8^54? (a) 6 (b) 4 (c) 8 (d) 2
6. 6.What is the units digit of 2^55? (a) 8 (b) 4 (c) 2 (d) 6
7. 7.Which of these numbers is divisible by 11? (a) 9657 (b) 9668 (c) 9676 (d) 9658
8. 8.What is the remainder when 248 is divided by 9? (a) 4 (b) 6 (c) 5 (d) 7
9. 9.Which of these numbers is divisible by 4? (a) 3120 (b) 3117 (c) 3121 (d) 3127
10. 10.What is the remainder when 708 is divided by 9? (a) 3 (b) 7 (c) 5 (d) 6
11. 11.Which of these numbers is divisible by 11? (a) 4310 (b) 4312 (c) 4324 (d) 4319
12. 12.What is the remainder when 263 is divided by 12? (a) 11 (b) 10 (c) 1 (d) 9
13. 13.What is the smallest number you must add to 279 to make it divisible by 10? (a) 3 (b) 9 (c) 2 (d) 1
14. 14.What is the smallest number you must add to 636 to make it divisible by 10? (a) 3 (b) 4 (c) 5 (d) 6
15. 15.What is the remainder when 587 is divided by 9? (a) 3 (b) 7 (c) 1 (d) 2
16. 16.Which of these numbers is divisible by 9? (a) 2216 (b) 2223 (c) 2230 (d) 2234
17. 17.What is the smallest number you must add to 310 to make it divisible by 12? (a) 3 (b) 10 (c) 1 (d) 2
18. 18.What is the units digit of 3^41? (a) 9 (b) 1 (c) 7 (d) 3
19. 19.Which of these numbers is divisible by 9? (a) 1983 (b) 1963 (c) 1973 (d) 1971
20. 20.What is the smallest number you must add to 559 to make it divisible by 7? (a) 2 (b) 6 (c) 1 (d) 3
21. 21.What is the smallest number you must add to 435 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 1
22. 22.What is the units digit of 4^32? (a) 5 (b) 7 (c) 6 (d) 4
23. 23.What is the units digit of 4^60? (a) 7 (b) 4 (c) 5 (d) 6
24. 24.What is the units digit of 9^44? (a) 0 (b) 2 (c) 1 (d) 9
25. 25.What is the smallest number you must add to 67 to make it divisible by 6? (a) 4 (b) 5 (c) 1 (d) 3
26. 26.Which of these numbers is divisible by 3? (a) 197 (b) 203 (c) 198 (d) 199
27. 27.What is the remainder when 635 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 2
28. 28.Which of these numbers is divisible by 11? (a) 7265 (b) 7244 (c) 7255 (d) 7249
29. 29.Which of these numbers is divisible by 11? (a) 2350 (b) 2359 (c) 2354 (d) 2374
30. 30.What is the remainder when 286 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
31. 31.What is the units digit of 8^22? (a) 4 (b) 6 (c) 8 (d) 2
32. 32.What is the remainder when 339 is divided by 7? (a) 3 (b) 2 (c) 4 (d) 5
33. 33.Which of these numbers is divisible by 3? (a) 2429 (b) 2426 (c) 2427 (d) 2432
34. 34.What is the smallest number you must add to 574 to make it divisible by 10? (a) 7 (b) 5 (c) 4 (d) 6
35. 35.What is the remainder when 68 is divided by 3? (a) 1 (b) 0 (c) 2
36. 36.What is the remainder when 65 is divided by 3? (a) 1 (b) 0 (c) 2
37. 37.What is the remainder when 696 is divided by 9? (a) 2 (b) 3 (c) 6 (d) 4
38. 38.Which of these numbers is divisible by 9? (a) 2118 (b) 2106 (c) 2099 (d) 2109
39. 39.Which of these numbers is divisible by 4? (a) 931 (b) 925 (c) 921 (d) 924
40. 40.What is the smallest number you must add to 414 to make it divisible by 10? (a) 5 (b) 4 (c) 7 (d) 6

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