Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 582 · 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 456 is divided by 11? (a) 6 (b) 4 (c) 7 (d) 5
2. 2.Which of these numbers is divisible by 4? (a) 988 (b) 995 (c) 991 (d) 985
3. 3.What is the units digit of 2^14? (a) 2 (b) 6 (c) 4 (d) 8
4. 4.What is the remainder when 899 is divided by 12? (a) 10 (b) 1 (c) 9 (d) 11
5. 5.What is the smallest number you must add to 606 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 6
6. 6.What is the units digit of 8^25? (a) 6 (b) 4 (c) 8 (d) 2
7. 7.What is the units digit of 3^53? (a) 9 (b) 1 (c) 3 (d) 7
8. 8.What is the smallest number you must add to 110 to make it divisible by 4? (a) 2 (b) 3 (c) 1
9. 9.What is the units digit of 3^38? (a) 7 (b) 1 (c) 9 (d) 3
10. 10.What is the units digit of 7^46? (a) 3 (b) 9 (c) 1 (d) 7
11. 11.What is the units digit of 4^24? (a) 7 (b) 5 (c) 6 (d) 4
12. 12.What is the units digit of 8^60? (a) 4 (b) 2 (c) 6 (d) 8
13. 13.What is the smallest number you must add to 304 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 7
14. 14.What is the units digit of 9^35? (a) 9 (b) 1 (c) 7 (d) 8
15. 15.What is the remainder when 103 is divided by 3? (a) 0 (b) 1 (c) 2
16. 16.What is the remainder when 212 is divided by 6? (a) 2 (b) 1 (c) 4 (d) 3
17. 17.What is the smallest number you must add to 221 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
18. 18.What is the remainder when 91 is divided by 9? (a) 0 (b) 1 (c) 2 (d) 8
19. 19.What is the remainder when 451 is divided by 6? (a) 0 (b) 5 (c) 2 (d) 1
20. 20.What is the remainder when 707 is divided by 9? (a) 6 (b) 7 (c) 5 (d) 4
21. 21.What is the remainder when 253 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
22. 22.What is the units digit of 2^48? (a) 6 (b) 4 (c) 2 (d) 8
23. 23.What is the smallest number you must add to 645 to make it divisible by 10? (a) 6 (b) 5 (c) 4 (d) 7
24. 24.What is the units digit of 4^49? (a) 5 (b) 4 (c) 6 (d) 3
25. 25.What is the smallest number you must add to 487 to make it divisible by 11? (a) 8 (b) 7 (c) 9 (d) 3
26. 26.Which of these numbers is divisible by 3? (a) 377 (b) 379 (c) 378 (d) 382
27. 27.What is the smallest number you must add to 325 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
28. 28.Which of these numbers is divisible by 11? (a) 8772 (b) 8767 (c) 8787 (d) 8761
29. 29.What is the units digit of 7^51? (a) 9 (b) 1 (c) 7 (d) 3
30. 30.Which of these numbers is divisible by 4? (a) 1618 (b) 1620 (c) 1625 (d) 1622
31. 31.What is the remainder when 172 is divided by 3? (a) 0 (b) 1 (c) 2
32. 32.What is the remainder when 83 is divided by 3? (a) 1 (b) 2 (c) 0
33. 33.Which of these numbers is divisible by 11? (a) 1642 (b) 1650 (c) 1671 (d) 1660
34. 34.What is the remainder when 76 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
35. 35.What is the smallest number you must add to 251 to make it divisible by 12? (a) 1 (b) 3 (c) 2 (d) 11
36. 36.What is the units digit of 9^27? (a) 9 (b) 1 (c) 8 (d) 7
37. 37.What is the remainder when 170 is divided by 12? (a) 3 (b) 2 (c) 1 (d) 10
38. 38.What is the remainder when 436 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
39. 39.What is the units digit of 3^55? (a) 1 (b) 9 (c) 3 (d) 7
40. 40.Which of these numbers is divisible by 4? (a) 1931 (b) 1928 (c) 1933 (d) 1927

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