Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 842 · 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 units digit of 9^46? (a) 2 (b) 9 (c) 0 (d) 1
2. 2.Which of these numbers is divisible by 11? (a) 7112 (b) 7102 (c) 7095 (d) 7087
3. 3.What is the units digit of 3^54? (a) 9 (b) 7 (c) 1 (d) 3
4. 4.What is the smallest number you must add to 587 to make it divisible by 11? (a) 7 (b) 4 (c) 6 (d) 8
5. 5.What is the remainder when 245 is divided by 10? (a) 5 (b) 4 (c) 7 (d) 6
6. 6.What is the units digit of 9^18? (a) 9 (b) 1 (c) 2 (d) 0
7. 7.Which of these numbers is divisible by 3? (a) 332 (b) 334 (c) 337 (d) 333
8. 8.What is the units digit of 2^58? (a) 2 (b) 4 (c) 8 (d) 6
9. 9.What is the remainder when 195 is divided by 11? (a) 9 (b) 8 (c) 3 (d) 7
10. 10.What is the smallest number you must add to 326 to make it divisible by 11? (a) 4 (b) 3 (c) 5 (d) 7
11. 11.What is the units digit of 8^40? (a) 6 (b) 8 (c) 4 (d) 2
12. 12.What is the smallest number you must add to 297 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
13. 13.Which of these numbers is divisible by 11? (a) 2765 (b) 2782 (c) 2772 (d) 2788
14. 14.Which of these numbers is divisible by 11? (a) 7977 (b) 8003 (c) 7986 (d) 7991
15. 15.Which of these numbers is divisible by 3? (a) 1300 (b) 1307 (c) 1304 (d) 1302
16. 16.Which of these numbers is divisible by 4? (a) 811 (b) 815 (c) 808 (d) 805
17. 17.What is the smallest number you must add to 491 to make it divisible by 8? (a) 4 (b) 3 (c) 6 (d) 5
18. 18.Which of these numbers is divisible by 11? (a) 6951 (b) 6958 (c) 6952 (d) 6966
19. 19.Which of these numbers is divisible by 4? (a) 1381 (b) 1376 (c) 1377 (d) 1375
20. 20.Which of these numbers is divisible by 8? (a) 445 (b) 455 (c) 458 (d) 448
21. 21.What is the remainder when 85 is divided by 3? (a) 1 (b) 2 (c) 0
22. 22.Which of these numbers is divisible by 9? (a) 6770 (b) 6759 (c) 6757 (d) 6762
23. 23.What is the units digit of 2^25? (a) 6 (b) 2 (c) 8 (d) 4
24. 24.Which of these numbers is divisible by 3? (a) 1615 (b) 1617 (c) 1622 (d) 1618
25. 25.What is the units digit of 3^48? (a) 7 (b) 1 (c) 3 (d) 9
26. 26.What is the units digit of 7^13? (a) 1 (b) 9 (c) 7 (d) 3
27. 27.What is the units digit of 2^44? (a) 2 (b) 8 (c) 4 (d) 6
28. 28.Which of these numbers is divisible by 6? (a) 2619 (b) 2616 (c) 2625 (d) 2614
29. 29.What is the remainder when 219 is divided by 11? (a) 8 (b) 9 (c) 10 (d) 1
30. 30.What is the smallest number you must add to 491 to make it divisible by 10? (a) 9 (b) 8 (c) 7 (d) 1
31. 31.Which of these numbers is divisible by 4? (a) 1521 (b) 1517 (c) 1516 (d) 1514
32. 32.What is the remainder when 242 is divided by 6? (a) 1 (b) 3 (c) 2 (d) 4
33. 33.What is the units digit of 2^41? (a) 4 (b) 2 (c) 6 (d) 8
34. 34.What is the smallest number you must add to 274 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
35. 35.What is the smallest number you must add to 464 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
36. 36.Which of these numbers is divisible by 6? (a) 3455 (b) 3446 (c) 3458 (d) 3450
37. 37.What is the remainder when 237 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
38. 38.What is the smallest number you must add to 785 to make it divisible by 11? (a) 6 (b) 7 (c) 4 (d) 8
39. 39.What is the units digit of 9^23? (a) 9 (b) 8 (c) 1 (d) 7
40. 40.What is the smallest number you must add to 151 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3

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