Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 173 · 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 230 is divided by 12? (a) 2 (b) 1 (c) 3 (d) 10
2. 2.What is the units digit of 2^24? (a) 8 (b) 4 (c) 2 (d) 6
3. 3.Which of these numbers is divisible by 3? (a) 436 (b) 438 (c) 442 (d) 439
4. 4.What is the remainder when 491 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
5. 5.What is the units digit of 9^52? (a) 9 (b) 2 (c) 0 (d) 1
6. 6.Which of these numbers is divisible by 3? (a) 205 (b) 203 (c) 204 (d) 209
7. 7.What is the smallest number you must add to 262 to make it divisible by 4? (a) 3 (b) 2 (c) 1
8. 8.Which of these numbers is divisible by 9? (a) 2956 (b) 2974 (c) 2961 (d) 2963
9. 9.What is the smallest number you must add to 124 to make it divisible by 11? (a) 3 (b) 9 (c) 7 (d) 8
10. 10.Which of these numbers is divisible by 4? (a) 2139 (b) 2135 (c) 2132 (d) 2130
11. 11.What is the units digit of 3^42? (a) 7 (b) 9 (c) 3 (d) 1
12. 12.What is the smallest number you must add to 554 to make it divisible by 12? (a) 9 (b) 2 (c) 10 (d) 11
13. 13.What is the units digit of 8^29? (a) 2 (b) 6 (c) 8 (d) 4
14. 14.Which of these numbers is divisible by 4? (a) 533 (b) 541 (c) 536 (d) 538
15. 15.What is the smallest number you must add to 303 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
16. 16.What is the units digit of 7^60? (a) 3 (b) 9 (c) 7 (d) 1
17. 17.What is the units digit of 9^27? (a) 7 (b) 9 (c) 8 (d) 1
18. 18.What is the smallest number you must add to 205 to make it divisible by 11? (a) 7 (b) 5 (c) 4 (d) 3
19. 19.What is the smallest number you must add to 231 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
20. 20.What is the remainder when 377 is divided by 9? (a) 1 (b) 6 (c) 8 (d) 7
21. 21.What is the units digit of 7^12? (a) 3 (b) 1 (c) 9 (d) 7
22. 22.What is the remainder when 129 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
23. 23.What is the smallest number you must add to 371 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
24. 24.Which of these numbers is divisible by 6? (a) 1609 (b) 1616 (c) 1608 (d) 1603
25. 25.What is the smallest number you must add to 669 to make it divisible by 10? (a) 9 (b) 2 (c) 3 (d) 1
26. 26.What is the units digit of 7^33? (a) 3 (b) 9 (c) 7 (d) 1
27. 27.Which of these numbers is divisible by 6? (a) 1417 (b) 1423 (c) 1411 (d) 1416
28. 28.What is the remainder when 234 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
29. 29.What is the remainder when 646 is divided by 11? (a) 3 (b) 9 (c) 7 (d) 8
30. 30.Which of these numbers is divisible by 6? (a) 4234 (b) 4236 (c) 4240 (d) 4246
31. 31.What is the units digit of 3^25? (a) 3 (b) 1 (c) 7 (d) 9
32. 32.What is the units digit of 4^28? (a) 7 (b) 5 (c) 6 (d) 4
33. 33.What is the units digit of 8^24? (a) 8 (b) 4 (c) 2 (d) 6
34. 34.Which of these numbers is divisible by 6? (a) 3586 (b) 3593 (c) 3581 (d) 3582
35. 35.What is the smallest number you must add to 232 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 4
36. 36.What is the smallest number you must add to 35 to make it divisible by 3? (a) 2 (b) 1
37. 37.Which of these numbers is divisible by 6? (a) 1451 (b) 1438 (c) 1445 (d) 1440
38. 38.What is the remainder when 61 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
39. 39.What is the remainder when 51 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4
40. 40.What is the smallest number you must add to 163 to make it divisible by 4? (a) 2 (b) 1 (c) 3

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