Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 121 · 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 244 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 2
2. 2.What is the units digit of 8^38? (a) 4 (b) 2 (c) 8 (d) 6
3. 3.What is the smallest number you must add to 379 to make it divisible by 12? (a) 4 (b) 6 (c) 7 (d) 5
4. 4.What is the smallest number you must add to 198 to make it divisible by 4? (a) 1 (b) 3 (c) 2
5. 5.What is the units digit of 2^45? (a) 2 (b) 4 (c) 6 (d) 8
6. 6.What is the smallest number you must add to 199 to make it divisible by 8? (a) 3 (b) 7 (c) 1 (d) 2
7. 7.What is the smallest number you must add to 46 to make it divisible by 3? (a) 1 (b) 2
8. 8.What is the smallest number you must add to 272 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
9. 9.What is the units digit of 7^29? (a) 7 (b) 1 (c) 3 (d) 9
10. 10.Which of these numbers is divisible by 9? (a) 6428 (b) 6435 (c) 6448 (d) 6441
11. 11.What is the remainder when 190 is divided by 9? (a) 0 (b) 1 (c) 8 (d) 2
12. 12.What is the smallest number you must add to 302 to make it divisible by 10? (a) 2 (b) 8 (c) 7 (d) 9
13. 13.What is the smallest number you must add to 198 to make it divisible by 8? (a) 1 (b) 6 (c) 2 (d) 3
14. 14.Which of these numbers is divisible by 9? (a) 7623 (b) 7633 (c) 7626 (d) 7615
15. 15.Which of these numbers is divisible by 6? (a) 3883 (b) 3890 (c) 3882 (d) 3877
16. 16.Which of these numbers is divisible by 3? (a) 1618 (b) 1617 (c) 1621 (d) 1615
17. 17.Which of these numbers is divisible by 3? (a) 1399 (b) 1403 (c) 1406 (d) 1401
18. 18.What is the units digit of 4^52? (a) 7 (b) 5 (c) 4 (d) 6
19. 19.Which of these numbers is divisible by 11? (a) 682 (b) 701 (c) 679 (d) 689
20. 20.What is the units digit of 2^15? (a) 8 (b) 6 (c) 2 (d) 4
21. 21.What is the remainder when 612 is divided by 11? (a) 6 (b) 8 (c) 4 (d) 7
22. 22.What is the remainder when 389 is divided by 9? (a) 2 (b) 3 (c) 1 (d) 7
23. 23.What is the remainder when 163 is divided by 6? (a) 0 (b) 1 (c) 5 (d) 2
24. 24.Which of these numbers is divisible by 6? (a) 876 (b) 875 (c) 883 (d) 877
25. 25.Which of these numbers is divisible by 9? (a) 1512 (b) 1515 (c) 1505 (d) 1529
26. 26.Which of these numbers is divisible by 6? (a) 2448 (b) 2450 (c) 2456 (d) 2445
27. 27.What is the units digit of 4^43? (a) 6 (b) 3 (c) 5 (d) 4
28. 28.Which of these numbers is divisible by 8? (a) 868 (b) 856 (c) 863 (d) 851
29. 29.What is the remainder when 161 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
30. 30.What is the units digit of 7^22? (a) 3 (b) 9 (c) 1 (d) 7
31. 31.What is the remainder when 426 is divided by 11? (a) 3 (b) 7 (c) 8 (d) 9
32. 32.What is the smallest number you must add to 47 to make it divisible by 3? (a) 2 (b) 1
33. 33.What is the smallest number you must add to 179 to make it divisible by 3? (a) 2 (b) 1
34. 34.Which of these numbers is divisible by 3? (a) 1152 (b) 1157 (c) 1150 (d) 1153
35. 35.What is the units digit of 8^20? (a) 8 (b) 6 (c) 2 (d) 4
36. 36.What is the units digit of 4^46? (a) 5 (b) 4 (c) 6 (d) 7
37. 37.Which of these numbers is divisible by 4? (a) 2966 (b) 2960 (c) 2962 (d) 2957
38. 38.What is the units digit of 4^35? (a) 6 (b) 3 (c) 4 (d) 5
39. 39.What is the smallest number you must add to 276 to make it divisible by 9? (a) 6 (b) 4 (c) 3 (d) 2
40. 40.What is the smallest number you must add to 377 to make it divisible by 8? (a) 5 (b) 7 (c) 1 (d) 6

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