Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 978 · 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 126 is divided by 8? (a) 2 (b) 7 (c) 5 (d) 6
2. 2.What is the smallest number you must add to 184 to make it divisible by 9? (a) 6 (b) 7 (c) 5 (d) 4
3. 3.What is the smallest number you must add to 468 to make it divisible by 7? (a) 6 (b) 1 (c) 3 (d) 2
4. 4.What is the smallest number you must add to 690 to make it divisible by 11? (a) 8 (b) 4 (c) 2 (d) 3
5. 5.What is the remainder when 51 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
6. 6.What is the smallest number you must add to 644 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
7. 7.Which of these numbers is divisible by 9? (a) 3733 (b) 3739 (c) 3735 (d) 3752
8. 8.What is the remainder when 398 is divided by 11? (a) 3 (b) 2 (c) 9 (d) 1
9. 9.What is the units digit of 7^33? (a) 9 (b) 3 (c) 1 (d) 7
10. 10.Which of these numbers is divisible by 3? (a) 1992 (b) 1991 (c) 1996 (d) 1993
11. 11.Which of these numbers is divisible by 11? (a) 3163 (b) 3170 (c) 3155 (d) 3157
12. 12.What is the units digit of 3^17? (a) 9 (b) 3 (c) 7 (d) 1
13. 13.What is the units digit of 2^53? (a) 8 (b) 2 (c) 4 (d) 6
14. 14.What is the remainder when 371 is divided by 11? (a) 7 (b) 3 (c) 8 (d) 9
15. 15.Which of these numbers is divisible by 8? (a) 3972 (b) 3968 (c) 3966 (d) 3980
16. 16.What is the units digit of 7^16? (a) 3 (b) 7 (c) 9 (d) 1
17. 17.What is the smallest number you must add to 62 to make it divisible by 4? (a) 3 (b) 1 (c) 2
18. 18.Which of these numbers is divisible by 4? (a) 3050 (b) 3045 (c) 3044 (d) 3043
19. 19.What is the smallest number you must add to 225 to make it divisible by 8? (a) 1 (b) 5 (c) 7 (d) 6
20. 20.What is the smallest number you must add to 185 to make it divisible by 4? (a) 1 (b) 2 (c) 3
21. 21.Which of these numbers is divisible by 3? (a) 2377 (b) 2375 (c) 2373 (d) 2372
22. 22.What is the smallest number you must add to 241 to make it divisible by 3? (a) 1 (b) 2
23. 23.Which of these numbers is divisible by 9? (a) 1669 (b) 1688 (c) 1674 (d) 1675
24. 24.Which of these numbers is divisible by 4? (a) 2480 (b) 2479 (c) 2487 (d) 2482
25. 25.What is the remainder when 915 is divided by 12? (a) 2 (b) 3 (c) 9 (d) 4
26. 26.What is the smallest number you must add to 73 to make it divisible by 3? (a) 1 (b) 2
27. 27.What is the remainder when 134 is divided by 3? (a) 1 (b) 0 (c) 2
28. 28.What is the remainder when 100 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
29. 29.What is the remainder when 318 is divided by 9? (a) 6 (b) 4 (c) 3 (d) 2
30. 30.What is the remainder when 476 is divided by 6? (a) 1 (b) 4 (c) 2 (d) 3
31. 31.What is the smallest number you must add to 727 to make it divisible by 12? (a) 4 (b) 5 (c) 6 (d) 7
32. 32.What is the smallest number you must add to 165 to make it divisible by 8? (a) 4 (b) 3 (c) 2 (d) 5
33. 33.What is the remainder when 261 is divided by 10? (a) 0 (b) 2 (c) 9 (d) 1
34. 34.What is the units digit of 2^58? (a) 2 (b) 8 (c) 4 (d) 6
35. 35.What is the smallest number you must add to 292 to make it divisible by 7? (a) 1 (b) 5 (c) 2 (d) 3
36. 36.What is the units digit of 8^22? (a) 6 (b) 2 (c) 4 (d) 8
37. 37.What is the units digit of 3^41? (a) 1 (b) 7 (c) 9 (d) 3
38. 38.What is the units digit of 9^47? (a) 8 (b) 7 (c) 9 (d) 1
39. 39.What is the units digit of 3^28? (a) 7 (b) 9 (c) 1 (d) 3
40. 40.What is the remainder when 274 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3

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