Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 457 · 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 868 to make it divisible by 12? (a) 9 (b) 8 (c) 7 (d) 4
2. 2.What is the smallest number you must add to 293 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
3. 3.What is the units digit of 2^38? (a) 4 (b) 2 (c) 6 (d) 8
4. 4.Which of these numbers is divisible by 9? (a) 3036 (b) 3048 (c) 3042 (d) 3054
5. 5.What is the units digit of 2^18? (a) 4 (b) 2 (c) 8 (d) 6
6. 6.Which of these numbers is divisible by 3? (a) 521 (b) 522 (c) 524 (d) 526
7. 7.Which of these numbers is divisible by 3? (a) 1307 (b) 1305 (c) 1304 (d) 1310
8. 8.What is the units digit of 7^27? (a) 1 (b) 9 (c) 7 (d) 3
9. 9.What is the units digit of 4^54? (a) 4 (b) 6 (c) 7 (d) 5
10. 10.Which of these numbers is divisible by 6? (a) 2862 (b) 2866 (c) 2869 (d) 2860
11. 11.Which of these numbers is divisible by 8? (a) 2184 (b) 2179 (c) 2199 (d) 2191
12. 12.What is the smallest number you must add to 162 to make it divisible by 4? (a) 2 (b) 3 (c) 1
13. 13.What is the smallest number you must add to 293 to make it divisible by 4? (a) 3 (b) 1 (c) 2
14. 14.What is the remainder when 218 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
15. 15.What is the units digit of 7^38? (a) 7 (b) 3 (c) 9 (d) 1
16. 16.What is the smallest number you must add to 198 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
17. 17.Which of these numbers is divisible by 11? (a) 1216 (b) 1210 (c) 1207 (d) 1222
18. 18.What is the units digit of 3^22? (a) 3 (b) 1 (c) 9 (d) 7
19. 19.What is the units digit of 2^45? (a) 8 (b) 4 (c) 2 (d) 6
20. 20.What is the units digit of 3^35? (a) 9 (b) 3 (c) 7 (d) 1
21. 21.What is the units digit of 3^46? (a) 7 (b) 9 (c) 1 (d) 3
22. 22.What is the remainder when 109 is divided by 3? (a) 2 (b) 0 (c) 1
23. 23.Which of these numbers is divisible by 6? (a) 4266 (b) 4275 (c) 4270 (d) 4261
24. 24.Which of these numbers is divisible by 4? (a) 3584 (b) 3581 (c) 3589 (d) 3587
25. 25.Which of these numbers is divisible by 3? (a) 1681 (b) 1685 (c) 1683 (d) 1687
26. 26.What is the units digit of 4^33? (a) 4 (b) 6 (c) 5 (d) 3
27. 27.What is the smallest number you must add to 396 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 3
28. 28.Which of these numbers is divisible by 6? (a) 922 (b) 927 (c) 915 (d) 918
29. 29.What is the smallest number you must add to 469 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
30. 30.What is the units digit of 4^46? (a) 5 (b) 7 (c) 6 (d) 4
31. 31.What is the remainder when 699 is divided by 12? (a) 4 (b) 9 (c) 2 (d) 3
32. 32.Which of these numbers is divisible by 4? (a) 2159 (b) 2152 (c) 2155 (d) 2149
33. 33.What is the smallest number you must add to 650 to make it divisible by 11? (a) 9 (b) 8 (c) 10 (d) 1
34. 34.What is the remainder when 273 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
35. 35.Which of these numbers is divisible by 6? (a) 4414 (b) 4401 (c) 4404 (d) 4405
36. 36.Which of these numbers is divisible by 9? (a) 7983 (b) 7984 (c) 7982 (d) 7997
37. 37.What is the units digit of 8^41? (a) 4 (b) 2 (c) 8 (d) 6
38. 38.What is the units digit of 2^19? (a) 4 (b) 2 (c) 8 (d) 6
39. 39.Which of these numbers is divisible by 4? (a) 1838 (b) 1840 (c) 1841 (d) 1845
40. 40.What is the remainder when 653 is divided by 10? (a) 2 (b) 4 (c) 3 (d) 7

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