Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 88 · 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 93 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
2. 2.What is the remainder when 276 is divided by 11? (a) 2 (b) 10 (c) 0 (d) 1
3. 3.What is the units digit of 9^27? (a) 1 (b) 8 (c) 9 (d) 7
4. 4.What is the smallest number you must add to 427 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 1
5. 5.What is the smallest number you must add to 246 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
6. 6.What is the units digit of 7^14? (a) 7 (b) 9 (c) 3 (d) 1
7. 7.Which of these numbers is divisible by 11? (a) 6433 (b) 6454 (c) 6435 (d) 6443
8. 8.Which of these numbers is divisible by 9? (a) 6948 (b) 6947 (c) 6958 (d) 6955
9. 9.Which of these numbers is divisible by 6? (a) 1843 (b) 1831 (c) 1836 (d) 1837
10. 10.What is the smallest number you must add to 131 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
11. 11.What is the units digit of 3^58? (a) 7 (b) 9 (c) 3 (d) 1
12. 12.What is the smallest number you must add to 760 to make it divisible by 11? (a) 1 (b) 9 (c) 10 (d) 8
13. 13.What is the units digit of 3^30? (a) 7 (b) 9 (c) 3 (d) 1
14. 14.Which of these numbers is divisible by 9? (a) 5508 (b) 5501 (c) 5512 (d) 5519
15. 15.What is the units digit of 8^14? (a) 2 (b) 8 (c) 6 (d) 4
16. 16.What is the units digit of 2^43? (a) 6 (b) 8 (c) 2 (d) 4
17. 17.What is the units digit of 3^36? (a) 3 (b) 7 (c) 9 (d) 1
18. 18.Which of these numbers is divisible by 9? (a) 6852 (b) 6835 (c) 6846 (d) 6840
19. 19.What is the remainder when 670 is divided by 9? (a) 3 (b) 5 (c) 4 (d) 6
20. 20.Which of these numbers is divisible by 3? (a) 2028 (b) 2026 (c) 2033 (d) 2030
21. 21.Which of these numbers is divisible by 8? (a) 3744 (b) 3754 (c) 3749 (d) 3740
22. 22.Which of these numbers is divisible by 11? (a) 1641 (b) 1631 (c) 1639 (d) 1651
23. 23.What is the smallest number you must add to 121 to make it divisible by 4? (a) 2 (b) 1 (c) 3
24. 24.What is the units digit of 8^18? (a) 8 (b) 6 (c) 2 (d) 4
25. 25.What is the units digit of 9^51? (a) 1 (b) 7 (c) 9 (d) 8
26. 26.What is the remainder when 486 is divided by 8? (a) 2 (b) 6 (c) 5 (d) 7
27. 27.What is the units digit of 9^44? (a) 1 (b) 0 (c) 9 (d) 2
28. 28.Which of these numbers is divisible by 8? (a) 3588 (b) 3597 (c) 3581 (d) 3584
29. 29.What is the smallest number you must add to 217 to make it divisible by 6? (a) 4 (b) 1 (c) 5 (d) 3
30. 30.What is the units digit of 2^18? (a) 2 (b) 8 (c) 6 (d) 4
31. 31.What is the remainder when 283 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
32. 32.What is the remainder when 301 is divided by 5? (a) 4 (b) 2 (c) 0 (d) 1
33. 33.What is the units digit of 8^45? (a) 4 (b) 8 (c) 6 (d) 2
34. 34.Which of these numbers is divisible by 6? (a) 2704 (b) 2694 (c) 2692 (d) 2699
35. 35.What is the remainder when 323 is divided by 8? (a) 2 (b) 4 (c) 5 (d) 3
36. 36.What is the units digit of 9^11? (a) 1 (b) 8 (c) 7 (d) 9
37. 37.What is the smallest number you must add to 257 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 2
38. 38.What is the remainder when 677 is divided by 10? (a) 7 (b) 8 (c) 3 (d) 6
39. 39.What is the units digit of 2^35? (a) 6 (b) 4 (c) 2 (d) 8
40. 40.What is the smallest number you must add to 688 to make it divisible by 11? (a) 5 (b) 4 (c) 7 (d) 6

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