Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 93 · 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 8^40? (a) 2 (b) 8 (c) 6 (d) 4
2. 2.What is the units digit of 4^18? (a) 5 (b) 7 (c) 4 (d) 6
3. 3.What is the remainder when 308 is divided by 12? (a) 8 (b) 9 (c) 7 (d) 4
4. 4.What is the remainder when 361 is divided by 10? (a) 9 (b) 0 (c) 2 (d) 1
5. 5.What is the smallest number you must add to 493 to make it divisible by 9? (a) 1 (b) 3 (c) 2 (d) 7
6. 6.What is the units digit of 4^10? (a) 5 (b) 4 (c) 7 (d) 6
7. 7.What is the remainder when 274 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
8. 8.Which of these numbers is divisible by 4? (a) 1779 (b) 1780 (c) 1787 (d) 1781
9. 9.What is the units digit of 7^21? (a) 3 (b) 7 (c) 9 (d) 1
10. 10.What is the smallest number you must add to 421 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 1
11. 11.What is the remainder when 118 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
12. 12.What is the units digit of 3^39? (a) 3 (b) 7 (c) 1 (d) 9
13. 13.Which of these numbers is divisible by 3? (a) 1954 (b) 1956 (c) 1957 (d) 1960
14. 14.Which of these numbers is divisible by 6? (a) 1417 (b) 1410 (c) 1414 (d) 1406
15. 15.What is the remainder when 199 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1
16. 16.What is the units digit of 8^32? (a) 6 (b) 8 (c) 2 (d) 4
17. 17.What is the units digit of 9^21? (a) 7 (b) 9 (c) 8 (d) 1
18. 18.What is the smallest number you must add to 508 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
19. 19.What is the smallest number you must add to 221 to make it divisible by 4? (a) 2 (b) 3 (c) 1
20. 20.Which of these numbers is divisible by 9? (a) 834 (b) 825 (c) 819 (d) 816
21. 21.What is the units digit of 9^24? (a) 0 (b) 2 (c) 1 (d) 9
22. 22.Which of these numbers is divisible by 9? (a) 3242 (b) 3254 (c) 3240 (d) 3235
23. 23.What is the smallest number you must add to 93 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
24. 24.Which of these numbers is divisible by 4? (a) 583 (b) 576 (c) 577 (d) 573
25. 25.What is the smallest number you must add to 95 to make it divisible by 8? (a) 2 (b) 7 (c) 3 (d) 1
26. 26.What is the remainder when 613 is divided by 10? (a) 7 (b) 3 (c) 4 (d) 2
27. 27.What is the units digit of 7^10? (a) 3 (b) 7 (c) 9 (d) 1
28. 28.Which of these numbers is divisible by 4? (a) 2474 (b) 2468 (c) 2466 (d) 2470
29. 29.Which of these numbers is divisible by 6? (a) 2984 (b) 2991 (c) 2981 (d) 2982
30. 30.What is the remainder when 179 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
31. 31.Which of these numbers is divisible by 3? (a) 1936 (b) 1939 (c) 1938 (d) 1942
32. 32.What is the units digit of 9^50? (a) 9 (b) 1 (c) 0 (d) 2
33. 33.What is the units digit of 2^17? (a) 4 (b) 6 (c) 2 (d) 8
34. 34.What is the remainder when 556 is divided by 8? (a) 4 (b) 5 (c) 6 (d) 3
35. 35.Which of these numbers is divisible by 3? (a) 2262 (b) 2261 (c) 2264 (d) 2266
36. 36.What is the units digit of 4^49? (a) 5 (b) 4 (c) 3 (d) 6
37. 37.What is the remainder when 87 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
38. 38.Which of these numbers is divisible by 4? (a) 1746 (b) 1739 (c) 1740 (d) 1742
39. 39.Which of these numbers is divisible by 3? (a) 1995 (b) 2000 (c) 1993 (d) 1997
40. 40.Which of these numbers is divisible by 3? (a) 514 (b) 508 (c) 510 (d) 511

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