Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 410 · 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 remainder when 136 is divided by 3? (a) 0 (b) 2 (c) 1
2. 2.What is the smallest number you must add to 124 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
3. 3.Which of these numbers is divisible by 4? (a) 1261 (b) 1265 (c) 1260 (d) 1257
4. 4.What is the units digit of 7^26? (a) 3 (b) 9 (c) 7 (d) 1
5. 5.Which of these numbers is divisible by 6? (a) 3402 (b) 3412 (c) 3405 (d) 3400
6. 6.What is the units digit of 2^13? (a) 4 (b) 2 (c) 6 (d) 8
7. 7.What is the remainder when 262 is divided by 9? (a) 1 (b) 2 (c) 8 (d) 0
8. 8.Which of these numbers is divisible by 11? (a) 8265 (b) 8251 (c) 8274 (d) 8261
9. 9.Which of these numbers is divisible by 6? (a) 1045 (b) 1053 (c) 1040 (d) 1044
10. 10.Which of these numbers is divisible by 3? (a) 2642 (b) 2648 (c) 2643 (d) 2644
11. 11.What is the remainder when 207 is divided by 6? (a) 3 (b) 2 (c) 5 (d) 4
12. 12.What is the remainder when 561 is divided by 7? (a) 6 (b) 0 (c) 1 (d) 2
13. 13.What is the remainder when 124 is divided by 11? (a) 8 (b) 3 (c) 4 (d) 2
14. 14.What is the units digit of 3^39? (a) 7 (b) 9 (c) 1 (d) 3
15. 15.Which of these numbers is divisible by 11? (a) 3564 (b) 3556 (c) 3573 (d) 3578
16. 16.What is the smallest number you must add to 246 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
17. 17.Which of these numbers is divisible by 8? (a) 4797 (b) 4814 (c) 4800 (d) 4803
18. 18.What is the units digit of 7^47? (a) 1 (b) 7 (c) 3 (d) 9
19. 19.Which of these numbers is divisible by 3? (a) 2122 (b) 2121 (c) 2125 (d) 2120
20. 20.What is the smallest number you must add to 532 to make it divisible by 9? (a) 7 (b) 1 (c) 6 (d) 8
21. 21.Which of these numbers is divisible by 3? (a) 1533 (b) 1532 (c) 1537 (d) 1534
22. 22.What is the remainder when 111 is divided by 5? (a) 2 (b) 0 (c) 1 (d) 4
23. 23.What is the units digit of 7^57? (a) 9 (b) 3 (c) 1 (d) 7
24. 24.What is the smallest number you must add to 354 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
25. 25.Which of these numbers is divisible by 9? (a) 4536 (b) 4531 (c) 4542 (d) 4552
26. 26.What is the smallest number you must add to 727 to make it divisible by 10? (a) 4 (b) 2 (c) 3 (d) 7
27. 27.What is the smallest number you must add to 269 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
28. 28.What is the units digit of 9^17? (a) 9 (b) 7 (c) 1 (d) 8
29. 29.Which of these numbers is divisible by 8? (a) 4294 (b) 4297 (c) 4288 (d) 4286
30. 30.Which of these numbers is divisible by 8? (a) 1280 (b) 1291 (c) 1276 (d) 1282
31. 31.What is the smallest number you must add to 311 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
32. 32.What is the smallest number you must add to 384 to make it divisible by 10? (a) 4 (b) 6 (c) 5 (d) 7
33. 33.Which of these numbers is divisible by 11? (a) 7927 (b) 7952 (c) 7931 (d) 7936
34. 34.Which of these numbers is divisible by 3? (a) 391 (b) 395 (c) 397 (d) 393
35. 35.What is the units digit of 2^39? (a) 4 (b) 2 (c) 6 (d) 8
36. 36.What is the smallest number you must add to 409 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
37. 37.What is the units digit of 4^40? (a) 4 (b) 7 (c) 5 (d) 6
38. 38.What is the remainder when 216 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 4
39. 39.Which of these numbers is divisible by 6? (a) 5344 (b) 5351 (c) 5346 (d) 5354
40. 40.What is the units digit of 3^37? (a) 9 (b) 1 (c) 3 (d) 7

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