Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 553 · 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 4? (a) 1869 (b) 1864 (c) 1863 (d) 1867
2. 2.What is the units digit of 9^13? (a) 1 (b) 8 (c) 7 (d) 9
3. 3.What is the remainder when 62 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 1
4. 4.What is the remainder when 315 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
5. 5.What is the smallest number you must add to 345 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 5
6. 6.What is the remainder when 308 is divided by 6? (a) 2 (b) 4 (c) 1 (d) 3
7. 7.What is the units digit of 2^13? (a) 6 (b) 2 (c) 8 (d) 4
8. 8.Which of these numbers is divisible by 6? (a) 2189 (b) 2190 (c) 2198 (d) 2192
9. 9.What is the remainder when 136 is divided by 7? (a) 2 (b) 3 (c) 4 (d) 5
10. 10.What is the units digit of 7^43? (a) 3 (b) 9 (c) 1 (d) 7
11. 11.Which of these numbers is divisible by 8? (a) 2133 (b) 2122 (c) 2119 (d) 2120
12. 12.Which of these numbers is divisible by 3? (a) 2079 (b) 2084 (c) 2078 (d) 2080
13. 13.Which of these numbers is divisible by 9? (a) 5746 (b) 5751 (c) 5763 (d) 5753
14. 14.What is the remainder when 51 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
15. 15.What is the remainder when 214 is divided by 8? (a) 2 (b) 7 (c) 6 (d) 5
16. 16.Which of these numbers is divisible by 3? (a) 2264 (b) 2258 (c) 2261 (d) 2259
17. 17.What is the remainder when 107 is divided by 9? (a) 1 (b) 6 (c) 7 (d) 8
18. 18.What is the smallest number you must add to 373 to make it divisible by 8? (a) 2 (b) 3 (c) 5 (d) 4
19. 19.Which of these numbers is divisible by 9? (a) 5443 (b) 5423 (c) 5427 (d) 5429
20. 20.What is the smallest number you must add to 127 to make it divisible by 12? (a) 5 (b) 4 (c) 6 (d) 7
21. 21.What is the smallest number you must add to 909 to make it divisible by 12? (a) 3 (b) 2 (c) 4 (d) 9
22. 22.What is the units digit of 3^24? (a) 9 (b) 7 (c) 3 (d) 1
23. 23.What is the remainder when 228 is divided by 11? (a) 7 (b) 9 (c) 3 (d) 8
24. 24.What is the smallest number you must add to 168 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
25. 25.What is the units digit of 3^15? (a) 1 (b) 3 (c) 7 (d) 9
26. 26.What is the smallest number you must add to 111 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 2
27. 27.Which of these numbers is divisible by 3? (a) 388 (b) 387 (c) 386 (d) 392
28. 28.Which of these numbers is divisible by 3? (a) 1642 (b) 1639 (c) 1641 (d) 1645
29. 29.What is the remainder when 43 is divided by 3? (a) 0 (b) 1 (c) 2
30. 30.What is the smallest number you must add to 265 to make it divisible by 4? (a) 3 (b) 2 (c) 1
31. 31.Which of these numbers is divisible by 8? (a) 3228 (b) 3232 (c) 3238 (d) 3244
32. 32.What is the smallest number you must add to 421 to make it divisible by 12? (a) 10 (b) 11 (c) 1 (d) 9
33. 33.Which of these numbers is divisible by 3? (a) 2272 (b) 2269 (c) 2271 (d) 2276
34. 34.What is the smallest number you must add to 212 to make it divisible by 3? (a) 1 (b) 2
35. 35.Which of these numbers is divisible by 6? (a) 1134 (b) 1145 (c) 1130 (d) 1138
36. 36.What is the smallest number you must add to 334 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
37. 37.What is the units digit of 8^37? (a) 6 (b) 4 (c) 8 (d) 2
38. 38.What is the smallest number you must add to 890 to make it divisible by 11? (a) 10 (b) 2 (c) 1 (d) 3
39. 39.What is the units digit of 8^31? (a) 2 (b) 6 (c) 4 (d) 8
40. 40.What is the smallest number you must add to 351 to make it divisible by 10? (a) 1 (b) 9 (c) 7 (d) 8

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