Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 315 · 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 11? (a) 3894 (b) 3897 (c) 3914 (d) 3893
2. 2.What is the smallest number you must add to 121 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
3. 3.What is the remainder when 397 is divided by 8? (a) 3 (b) 5 (c) 6 (d) 4
4. 4.Which of these numbers is divisible by 9? (a) 6102 (b) 6097 (c) 6106 (d) 6112
5. 5.Which of these numbers is divisible by 4? (a) 1940 (b) 1937 (c) 1942 (d) 1947
6. 6.What is the remainder when 219 is divided by 7? (a) 5 (b) 1 (c) 3 (d) 2
7. 7.Which of these numbers is divisible by 4? (a) 2837 (b) 2841 (c) 2846 (d) 2840
8. 8.What is the units digit of 3^53? (a) 7 (b) 3 (c) 1 (d) 9
9. 9.What is the smallest number you must add to 142 to make it divisible by 6? (a) 4 (b) 2 (c) 1 (d) 3
10. 10.What is the smallest number you must add to 43 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the remainder when 272 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
12. 12.What is the units digit of 2^35? (a) 2 (b) 6 (c) 8 (d) 4
13. 13.What is the smallest number you must add to 555 to make it divisible by 12? (a) 10 (b) 8 (c) 9 (d) 3
14. 14.What is the units digit of 8^54? (a) 8 (b) 2 (c) 6 (d) 4
15. 15.Which of these numbers is divisible by 8? (a) 1492 (b) 1482 (c) 1474 (d) 1480
16. 16.Which of these numbers is divisible by 6? (a) 3076 (b) 3064 (c) 3066 (d) 3071
17. 17.Which of these numbers is divisible by 8? (a) 6853 (b) 6869 (c) 6859 (d) 6856
18. 18.What is the units digit of 9^60? (a) 0 (b) 1 (c) 9 (d) 2
19. 19.What is the remainder when 249 is divided by 8? (a) 1 (b) 0 (c) 7 (d) 2
20. 20.What is the remainder when 211 is divided by 3? (a) 2 (b) 1 (c) 0
21. 21.What is the units digit of 8^16? (a) 8 (b) 6 (c) 4 (d) 2
22. 22.What is the units digit of 8^48? (a) 2 (b) 8 (c) 6 (d) 4
23. 23.What is the remainder when 77 is divided by 3? (a) 2 (b) 0 (c) 1
24. 24.What is the units digit of 3^40? (a) 9 (b) 7 (c) 3 (d) 1
25. 25.Which of these numbers is divisible by 6? (a) 3956 (b) 3954 (c) 3951 (d) 3963
26. 26.What is the remainder when 184 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 2
27. 27.Which of these numbers is divisible by 11? (a) 2866 (b) 2851 (c) 2845 (d) 2849
28. 28.What is the smallest number you must add to 399 to make it divisible by 11? (a) 9 (b) 7 (c) 8 (d) 3
29. 29.What is the smallest number you must add to 583 to make it divisible by 12? (a) 4 (b) 6 (c) 5 (d) 7
30. 30.What is the remainder when 299 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 1
31. 31.Which of these numbers is divisible by 8? (a) 4581 (b) 4568 (c) 4561 (d) 4570
32. 32.Which of these numbers is divisible by 3? (a) 1648 (b) 1644 (c) 1642 (d) 1645
33. 33.What is the smallest number you must add to 160 to make it divisible by 3? (a) 2 (b) 1
34. 34.Which of these numbers is divisible by 3? (a) 1896 (b) 1901 (c) 1897 (d) 1894
35. 35.What is the smallest number you must add to 331 to make it divisible by 12? (a) 4 (b) 7 (c) 6 (d) 5
36. 36.What is the units digit of 2^46? (a) 2 (b) 8 (c) 6 (d) 4
37. 37.What is the remainder when 148 is divided by 3? (a) 1 (b) 2 (c) 0
38. 38.What is the smallest number you must add to 507 to make it divisible by 8? (a) 6 (b) 5 (c) 3 (d) 4
39. 39.What is the units digit of 2^60? (a) 8 (b) 2 (c) 4 (d) 6
40. 40.What is the smallest number you must add to 233 to make it divisible by 4? (a) 1 (b) 2 (c) 3

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