Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 111 · 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 52 is divided by 3? (a) 1 (b) 0 (c) 2
2. 2.Which of these numbers is divisible by 3? (a) 268 (b) 265 (c) 263 (d) 264
3. 3.Which of these numbers is divisible by 11? (a) 1540 (b) 1548 (c) 1559 (d) 1532
4. 4.What is the units digit of 4^12? (a) 6 (b) 4 (c) 7 (d) 5
5. 5.What is the smallest number you must add to 58 to make it divisible by 3? (a) 1 (b) 2
6. 6.What is the remainder when 466 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 2
7. 7.What is the smallest number you must add to 646 to make it divisible by 10? (a) 6 (b) 3 (c) 5 (d) 4
8. 8.What is the smallest number you must add to 140 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 4
9. 9.Which of these numbers is divisible by 3? (a) 2158 (b) 2161 (c) 2160 (d) 2165
10. 10.What is the smallest number you must add to 164 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
11. 11.What is the remainder when 505 is divided by 7? (a) 6 (b) 2 (c) 1 (d) 0
12. 12.What is the remainder when 378 is divided by 8? (a) 3 (b) 1 (c) 6 (d) 2
13. 13.What is the units digit of 3^41? (a) 1 (b) 3 (c) 9 (d) 7
14. 14.Which of these numbers is divisible by 8? (a) 2525 (b) 2530 (c) 2542 (d) 2528
15. 15.What is the units digit of 3^42? (a) 1 (b) 9 (c) 7 (d) 3
16. 16.What is the units digit of 2^45? (a) 4 (b) 6 (c) 8 (d) 2
17. 17.Which of these numbers is divisible by 4? (a) 1521 (b) 1520 (c) 1525 (d) 1519
18. 18.What is the remainder when 777 is divided by 11? (a) 7 (b) 8 (c) 4 (d) 6
19. 19.Which of these numbers is divisible by 3? (a) 2442 (b) 2440 (c) 2443 (d) 2447
20. 20.What is the remainder when 322 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
21. 21.What is the remainder when 298 is divided by 11? (a) 2 (b) 10 (c) 0 (d) 1
22. 22.What is the units digit of 2^55? (a) 4 (b) 2 (c) 8 (d) 6
23. 23.Which of these numbers is divisible by 8? (a) 613 (b) 596 (c) 600 (d) 603
24. 24.What is the smallest number you must add to 169 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
25. 25.What is the remainder when 778 is divided by 10? (a) 2 (b) 7 (c) 8 (d) 9
26. 26.What is the remainder when 162 is divided by 8? (a) 6 (b) 3 (c) 2 (d) 1
27. 27.What is the smallest number you must add to 125 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 1
28. 28.Which of these numbers is divisible by 4? (a) 3548 (b) 3553 (c) 3551 (d) 3547
29. 29.What is the smallest number you must add to 174 to make it divisible by 4? (a) 1 (b) 3 (c) 2
30. 30.What is the remainder when 230 is divided by 9? (a) 6 (b) 4 (c) 5 (d) 7
31. 31.Which of these numbers is divisible by 9? (a) 2367 (b) 2371 (c) 2379 (d) 2365
32. 32.What is the units digit of 4^42? (a) 5 (b) 7 (c) 6 (d) 4
33. 33.What is the units digit of 8^23? (a) 8 (b) 6 (c) 4 (d) 2
34. 34.What is the remainder when 286 is divided by 7? (a) 6 (b) 4 (c) 1 (d) 5
35. 35.What is the smallest number you must add to 265 to make it divisible by 12? (a) 1 (b) 11 (c) 9 (d) 10
36. 36.Which of these numbers is divisible by 8? (a) 3127 (b) 3128 (c) 3129 (d) 3143
37. 37.What is the remainder when 361 is divided by 8? (a) 2 (b) 7 (c) 0 (d) 1
38. 38.What is the smallest number you must add to 952 to make it divisible by 12? (a) 8 (b) 4 (c) 7 (d) 9
39. 39.What is the remainder when 437 is divided by 10? (a) 7 (b) 8 (c) 3 (d) 6
40. 40.What is the remainder when 465 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 5

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