Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 24 · 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 600 is divided by 11? (a) 7 (b) 5 (c) 8 (d) 6
2. 2.What is the units digit of 9^35? (a) 1 (b) 9 (c) 8 (d) 7
3. 3.What is the units digit of 4^31? (a) 3 (b) 5 (c) 6 (d) 4
4. 4.What is the smallest number you must add to 239 to make it divisible by 3? (a) 2 (b) 1
5. 5.What is the remainder when 325 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
6. 6.What is the remainder when 229 is divided by 6? (a) 2 (b) 0 (c) 5 (d) 1
7. 7.What is the units digit of 7^30? (a) 1 (b) 9 (c) 7 (d) 3
8. 8.What is the remainder when 266 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
9. 9.Which of these numbers is divisible by 3? (a) 1544 (b) 1540 (c) 1539 (d) 1537
10. 10.What is the units digit of 9^24? (a) 9 (b) 1 (c) 0 (d) 2
11. 11.What is the smallest number you must add to 203 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
12. 12.Which of these numbers is divisible by 3? (a) 2276 (b) 2273 (c) 2269 (d) 2271
13. 13.What is the remainder when 771 is divided by 10? (a) 9 (b) 1 (c) 2 (d) 0
14. 14.Which of these numbers is divisible by 3? (a) 1925 (b) 1918 (c) 1921 (d) 1920
15. 15.What is the smallest number you must add to 231 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
16. 16.What is the smallest number you must add to 473 to make it divisible by 9? (a) 6 (b) 3 (c) 4 (d) 5
17. 17.What is the remainder when 357 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
18. 18.Which of these numbers is divisible by 11? (a) 7056 (b) 7051 (c) 7050 (d) 7069
19. 19.What is the remainder when 63 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
20. 20.Which of these numbers is divisible by 8? (a) 5557 (b) 5542 (c) 5550 (d) 5544
21. 21.What is the smallest number you must add to 202 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
22. 22.What is the remainder when 239 is divided by 8? (a) 5 (b) 6 (c) 1 (d) 7
23. 23.What is the remainder when 403 is divided by 10? (a) 7 (b) 4 (c) 3 (d) 2
24. 24.Which of these numbers is divisible by 9? (a) 3430 (b) 3426 (c) 3446 (d) 3429
25. 25.What is the smallest number you must add to 324 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
26. 26.What is the remainder when 807 is divided by 11? (a) 7 (b) 5 (c) 3 (d) 4
27. 27.Which of these numbers is divisible by 9? (a) 2729 (b) 2752 (c) 2740 (d) 2736
28. 28.What is the smallest number you must add to 341 to make it divisible by 8? (a) 3 (b) 4 (c) 2 (d) 5
29. 29.What is the smallest number you must add to 229 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 1
30. 30.Which of these numbers is divisible by 11? (a) 6983 (b) 6990 (c) 7005 (d) 6985
31. 31.What is the smallest number you must add to 202 to make it divisible by 7? (a) 2 (b) 3 (c) 6 (d) 1
32. 32.What is the remainder when 224 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 2
33. 33.Which of these numbers is divisible by 3? (a) 819 (b) 824 (c) 817 (d) 821
34. 34.What is the remainder when 291 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
35. 35.What is the smallest number you must add to 390 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 6
36. 36.Which of these numbers is divisible by 3? (a) 1493 (b) 1495 (c) 1491 (d) 1489
37. 37.What is the remainder when 104 is divided by 3? (a) 2 (b) 1 (c) 0
38. 38.Which of these numbers is divisible by 8? (a) 4224 (b) 4229 (c) 4238 (d) 4223
39. 39.What is the remainder when 157 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
40. 40.What is the units digit of 4^39? (a) 5 (b) 4 (c) 3 (d) 6

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