Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 266 · 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 49 is divided by 3? (a) 1 (b) 0 (c) 2
2. 2.What is the smallest number you must add to 354 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
3. 3.What is the units digit of 4^25? (a) 4 (b) 5 (c) 3 (d) 6
4. 4.Which of these numbers is divisible by 6? (a) 1914 (b) 1921 (c) 1913 (d) 1917
5. 5.What is the smallest number you must add to 161 to make it divisible by 9? (a) 3 (b) 8 (c) 1 (d) 2
6. 6.What is the remainder when 313 is divided by 7? (a) 2 (b) 5 (c) 4 (d) 6
7. 7.What is the units digit of 2^35? (a) 2 (b) 8 (c) 4 (d) 6
8. 8.What is the remainder when 161 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
9. 9.What is the units digit of 7^16? (a) 1 (b) 9 (c) 7 (d) 3
10. 10.Which of these numbers is divisible by 6? (a) 3883 (b) 3888 (c) 3889 (d) 3895
11. 11.Which of these numbers is divisible by 8? (a) 5934 (b) 5936 (c) 5939 (d) 5946
12. 12.Which of these numbers is divisible by 11? (a) 1713 (b) 1705 (c) 1699 (d) 1724
13. 13.Which of these numbers is divisible by 6? (a) 1655 (b) 1650 (c) 1649 (d) 1658
14. 14.Which of these numbers is divisible by 3? (a) 2072 (b) 2069 (c) 2075 (d) 2070
15. 15.What is the smallest number you must add to 511 to make it divisible by 8? (a) 3 (b) 2 (c) 7 (d) 1
16. 16.What is the remainder when 215 is divided by 8? (a) 7 (b) 5 (c) 6 (d) 1
17. 17.Which of these numbers is divisible by 3? (a) 1943 (b) 1946 (c) 1948 (d) 1944
18. 18.What is the units digit of 7^10? (a) 1 (b) 9 (c) 3 (d) 7
19. 19.What is the smallest number you must add to 471 to make it divisible by 7? (a) 2 (b) 5 (c) 6 (d) 4
20. 20.What is the remainder when 277 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
21. 21.Which of these numbers is divisible by 4? (a) 192 (b) 191 (c) 193 (d) 198
22. 22.What is the units digit of 9^55? (a) 7 (b) 8 (c) 9 (d) 1
23. 23.What is the units digit of 9^30? (a) 9 (b) 2 (c) 1 (d) 0
24. 24.Which of these numbers is divisible by 4? (a) 2423 (b) 2426 (c) 2420 (d) 2418
25. 25.What is the remainder when 793 is divided by 11? (a) 0 (b) 10 (c) 2 (d) 1
26. 26.Which of these numbers is divisible by 11? (a) 5392 (b) 5369 (c) 5380 (d) 5379
27. 27.What is the units digit of 3^59? (a) 9 (b) 3 (c) 7 (d) 1
28. 28.What is the remainder when 211 is divided by 8? (a) 3 (b) 2 (c) 5 (d) 4
29. 29.What is the remainder when 442 is divided by 8? (a) 2 (b) 3 (c) 1 (d) 6
30. 30.Which of these numbers is divisible by 4? (a) 2296 (b) 2303 (c) 2294 (d) 2299
31. 31.What is the remainder when 584 is divided by 9? (a) 7 (b) 8 (c) 6 (d) 1
32. 32.Which of these numbers is divisible by 3? (a) 860 (b) 863 (c) 857 (d) 858
33. 33.What is the smallest number you must add to 73 to make it divisible by 3? (a) 1 (b) 2
34. 34.What is the remainder when 183 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
35. 35.Which of these numbers is divisible by 9? (a) 4840 (b) 4824 (c) 4826 (d) 4817
36. 36.What is the remainder when 102 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 3
37. 37.What is the remainder when 285 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
38. 38.What is the remainder when 189 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
39. 39.What is the units digit of 9^41? (a) 1 (b) 7 (c) 9 (d) 8
40. 40.What is the smallest number you must add to 591 to make it divisible by 11? (a) 4 (b) 8 (c) 3 (d) 2

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