Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 41 · 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.Which of these numbers is divisible by 8? (a) 5998 (b) 5987 (c) 5984 (d) 5981
2. 2.What is the smallest number you must add to 135 to make it divisible by 11? (a) 9 (b) 3 (c) 7 (d) 8
3. 3.What is the units digit of 7^36? (a) 7 (b) 3 (c) 1 (d) 9
4. 4.What is the smallest number you must add to 137 to make it divisible by 7? (a) 2 (b) 4 (c) 3 (d) 5
5. 5.What is the smallest number you must add to 221 to make it divisible by 4? (a) 2 (b) 3 (c) 1
6. 6.Which of these numbers is divisible by 6? (a) 1931 (b) 1932 (c) 1934 (d) 1940
7. 7.Which of these numbers is divisible by 6? (a) 3930 (b) 3926 (c) 3940 (d) 3931
8. 8.What is the smallest number you must add to 41 to make it divisible by 4? (a) 2 (b) 3 (c) 1
9. 9.What is the smallest number you must add to 135 to make it divisible by 12? (a) 3 (b) 8 (c) 10 (d) 9
10. 10.What is the remainder when 444 is divided by 10? (a) 3 (b) 5 (c) 6 (d) 4
11. 11.Which of these numbers is divisible by 4? (a) 1908 (b) 1914 (c) 1911 (d) 1906
12. 12.What is the units digit of 7^23? (a) 7 (b) 9 (c) 3 (d) 1
13. 13.What is the units digit of 3^32? (a) 7 (b) 1 (c) 3 (d) 9
14. 14.Which of these numbers is divisible by 8? (a) 2914 (b) 2899 (c) 2904 (d) 2907
15. 15.Which of these numbers is divisible by 6? (a) 1154 (b) 1149 (c) 1160 (d) 1152
16. 16.What is the smallest number you must add to 491 to make it divisible by 11? (a) 4 (b) 7 (c) 3 (d) 5
17. 17.What is the remainder when 332 is divided by 9? (a) 8 (b) 1 (c) 7 (d) 6
18. 18.What is the remainder when 372 is divided by 9? (a) 4 (b) 3 (c) 2 (d) 6
19. 19.Which of these numbers is divisible by 11? (a) 2812 (b) 2828 (c) 2823 (d) 2816
20. 20.What is the smallest number you must add to 109 to make it divisible by 7? (a) 4 (b) 3 (c) 5 (d) 2
21. 21.What is the smallest number you must add to 266 to make it divisible by 4? (a) 1 (b) 3 (c) 2
22. 22.What is the remainder when 97 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
23. 23.What is the units digit of 9^28? (a) 9 (b) 2 (c) 1 (d) 0
24. 24.Which of these numbers is divisible by 4? (a) 2448 (b) 2449 (c) 2453 (d) 2447
25. 25.What is the units digit of 4^51? (a) 4 (b) 6 (c) 5 (d) 3
26. 26.What is the units digit of 4^31? (a) 4 (b) 5 (c) 6 (d) 3
27. 27.Which of these numbers is divisible by 9? (a) 2244 (b) 2241 (c) 2239 (d) 2256
28. 28.Which of these numbers is divisible by 9? (a) 3555 (b) 3556 (c) 3571 (d) 3551
29. 29.Which of these numbers is divisible by 6? (a) 1654 (b) 1661 (c) 1650 (d) 1648
30. 30.Which of these numbers is divisible by 8? (a) 1780 (b) 1776 (c) 1769 (d) 1786
31. 31.What is the units digit of 9^52? (a) 0 (b) 9 (c) 2 (d) 1
32. 32.What is the remainder when 227 is divided by 12? (a) 11 (b) 10 (c) 9 (d) 1
33. 33.Which of these numbers is divisible by 4? (a) 3034 (b) 3026 (c) 3029 (d) 3028
34. 34.Which of these numbers is divisible by 11? (a) 7405 (b) 7414 (c) 7428 (d) 7419
35. 35.What is the remainder when 113 is divided by 3? (a) 2 (b) 0 (c) 1
36. 36.Which of these numbers is divisible by 9? (a) 6720 (b) 6714 (c) 6731 (d) 6709
37. 37.What is the units digit of 7^34? (a) 7 (b) 3 (c) 9 (d) 1
38. 38.What is the smallest number you must add to 95 to make it divisible by 8? (a) 2 (b) 3 (c) 7 (d) 1
39. 39.What is the smallest number you must add to 719 to make it divisible by 10? (a) 3 (b) 2 (c) 9 (d) 1
40. 40.Which of these numbers is divisible by 8? (a) 2957 (b) 2964 (c) 2960 (d) 2971

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