Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 779 · 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 smallest number you must add to 542 to make it divisible by 11? (a) 7 (b) 9 (c) 8 (d) 3
2. 2.What is the units digit of 3^24? (a) 9 (b) 3 (c) 1 (d) 7
3. 3.What is the remainder when 238 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
4. 4.What is the smallest number you must add to 871 to make it divisible by 11? (a) 9 (b) 2 (c) 8 (d) 10
5. 5.Which of these numbers is divisible by 4? (a) 2055 (b) 2047 (c) 2049 (d) 2048
6. 6.What is the smallest number you must add to 323 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 1
7. 7.What is the smallest number you must add to 57 to make it divisible by 4? (a) 2 (b) 1 (c) 3
8. 8.Which of these numbers is divisible by 8? (a) 1052 (b) 1058 (c) 1056 (d) 1070
9. 9.What is the smallest number you must add to 137 to make it divisible by 3? (a) 1 (b) 2
10. 10.Which of these numbers is divisible by 3? (a) 615 (b) 613 (c) 617 (d) 620
11. 11.What is the units digit of 4^50? (a) 5 (b) 7 (c) 4 (d) 6
12. 12.What is the units digit of 7^18? (a) 1 (b) 7 (c) 3 (d) 9
13. 13.Which of these numbers is divisible by 4? (a) 2762 (b) 2759 (c) 2760 (d) 2766
14. 14.Which of these numbers is divisible by 3? (a) 876 (b) 880 (c) 874 (d) 878
15. 15.Which of these numbers is divisible by 3? (a) 1826 (b) 1822 (c) 1824 (d) 1829
16. 16.Which of these numbers is divisible by 8? (a) 1935 (b) 1939 (c) 1936 (d) 1946
17. 17.What is the units digit of 7^23? (a) 1 (b) 3 (c) 9 (d) 7
18. 18.What is the units digit of 4^59? (a) 6 (b) 5 (c) 4 (d) 3
19. 19.What is the smallest number you must add to 424 to make it divisible by 10? (a) 5 (b) 4 (c) 7 (d) 6
20. 20.Which of these numbers is divisible by 8? (a) 4418 (b) 4409 (c) 4416 (d) 4425
21. 21.What is the remainder when 61 is divided by 3? (a) 0 (b) 1 (c) 2
22. 22.What is the smallest number you must add to 240 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 2
23. 23.What is the remainder when 371 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 1
24. 24.What is the remainder when 779 is divided by 10? (a) 9 (b) 1 (c) 7 (d) 8
25. 25.What is the units digit of 3^16? (a) 1 (b) 9 (c) 7 (d) 3
26. 26.What is the remainder when 239 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
27. 27.What is the remainder when 59 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
28. 28.What is the units digit of 9^59? (a) 9 (b) 7 (c) 1 (d) 8
29. 29.What is the smallest number you must add to 50 to make it divisible by 4? (a) 3 (b) 1 (c) 2
30. 30.Which of these numbers is divisible by 8? (a) 4193 (b) 4184 (c) 4177 (d) 4185
31. 31.Which of these numbers is divisible by 11? (a) 3667 (b) 3653 (c) 3652 (d) 3646
32. 32.Which of these numbers is divisible by 3? (a) 527 (b) 525 (c) 530 (d) 524
33. 33.Which of these numbers is divisible by 8? (a) 2953 (b) 2947 (c) 2962 (d) 2952
34. 34.Which of these numbers is divisible by 8? (a) 3954 (b) 3960 (c) 3972 (d) 3962
35. 35.Which of these numbers is divisible by 4? (a) 3164 (b) 3167 (c) 3170 (d) 3163
36. 36.What is the smallest number you must add to 170 to make it divisible by 4? (a) 1 (b) 2 (c) 3
37. 37.What is the remainder when 131 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 6
38. 38.What is the smallest number you must add to 143 to make it divisible by 10? (a) 3 (b) 8 (c) 7 (d) 6
39. 39.Which of these numbers is divisible by 6? (a) 1404 (b) 1407 (c) 1411 (d) 1400
40. 40.What is the remainder when 344 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1

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