Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 318 · 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 213 to make it divisible by 4? (a) 3 (b) 1 (c) 2
2. 2.Which of these numbers is divisible by 9? (a) 2864 (b) 2853 (c) 2849 (d) 2859
3. 3.What is the smallest number you must add to 306 to make it divisible by 4? (a) 2 (b) 3 (c) 1
4. 4.What is the smallest number you must add to 164 to make it divisible by 9? (a) 6 (b) 8 (c) 2 (d) 7
5. 5.What is the smallest number you must add to 162 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
6. 6.Which of these numbers is divisible by 4? (a) 2273 (b) 2276 (c) 2281 (d) 2277
7. 7.What is the units digit of 2^38? (a) 8 (b) 2 (c) 6 (d) 4
8. 8.Which of these numbers is divisible by 4? (a) 417 (b) 420 (c) 423 (d) 426
9. 9.What is the smallest number you must add to 314 to make it divisible by 4? (a) 3 (b) 2 (c) 1
10. 10.What is the units digit of 3^26? (a) 9 (b) 7 (c) 1 (d) 3
11. 11.What is the smallest number you must add to 51 to make it divisible by 4? (a) 3 (b) 1 (c) 2
12. 12.What is the remainder when 306 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
13. 13.Which of these numbers is divisible by 9? (a) 6361 (b) 6350 (c) 6345 (d) 6343
14. 14.What is the remainder when 265 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
15. 15.What is the units digit of 4^40? (a) 5 (b) 7 (c) 6 (d) 4
16. 16.Which of these numbers is divisible by 6? (a) 2119 (b) 2117 (c) 2128 (d) 2118
17. 17.Which of these numbers is divisible by 4? (a) 362 (b) 371 (c) 364 (d) 365
18. 18.What is the smallest number you must add to 71 to make it divisible by 3? (a) 2 (b) 1
19. 19.Which of these numbers is divisible by 11? (a) 8965 (b) 8961 (c) 8983 (d) 8967
20. 20.What is the remainder when 898 is divided by 12? (a) 11 (b) 10 (c) 2 (d) 9
21. 21.Which of these numbers is divisible by 4? (a) 411 (b) 413 (c) 405 (d) 408
22. 22.What is the units digit of 7^56? (a) 7 (b) 1 (c) 9 (d) 3
23. 23.What is the remainder when 356 is divided by 8? (a) 6 (b) 3 (c) 4 (d) 5
24. 24.Which of these numbers is divisible by 4? (a) 1098 (b) 1102 (c) 1100 (d) 1107
25. 25.What is the remainder when 149 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
26. 26.What is the remainder when 425 is divided by 8? (a) 0 (b) 1 (c) 2 (d) 7
27. 27.Which of these numbers is divisible by 9? (a) 3750 (b) 3753 (c) 3758 (d) 3765
28. 28.Which of these numbers is divisible by 9? (a) 2601 (b) 2594 (c) 2613 (d) 2603
29. 29.What is the smallest number you must add to 183 to make it divisible by 11? (a) 5 (b) 7 (c) 4 (d) 3
30. 30.Which of these numbers is divisible by 4? (a) 3525 (b) 3521 (c) 3520 (d) 3517
31. 31.What is the remainder when 802 is divided by 11? (a) 10 (b) 9 (c) 8 (d) 1
32. 32.What is the remainder when 81 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 6
33. 33.What is the remainder when 214 is divided by 8? (a) 2 (b) 6 (c) 7 (d) 5
34. 34.What is the units digit of 3^31? (a) 3 (b) 9 (c) 1 (d) 7
35. 35.What is the remainder when 546 is divided by 12? (a) 6 (b) 5 (c) 8 (d) 7
36. 36.Which of these numbers is divisible by 11? (a) 9900 (b) 9916 (c) 9907 (d) 9895
37. 37.Which of these numbers is divisible by 3? (a) 249 (b) 254 (c) 248 (d) 250
38. 38.What is the smallest number you must add to 363 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 5
39. 39.What is the smallest number you must add to 474 to make it divisible by 10? (a) 7 (b) 6 (c) 5 (d) 4
40. 40.What is the units digit of 4^37? (a) 3 (b) 5 (c) 4 (d) 6

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