Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 469 · 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 522 to make it divisible by 8? (a) 6 (b) 2 (c) 5 (d) 7
2. 2.Which of these numbers is divisible by 3? (a) 2285 (b) 2290 (c) 2288 (d) 2286
3. 3.What is the remainder when 697 is divided by 12? (a) 11 (b) 0 (c) 1 (d) 2
4. 4.What is the units digit of 4^42? (a) 7 (b) 5 (c) 4 (d) 6
5. 5.What is the units digit of 3^31? (a) 3 (b) 7 (c) 1 (d) 9
6. 6.Which of these numbers is divisible by 6? (a) 3917 (b) 3909 (c) 3906 (d) 3903
7. 7.Which of these numbers is divisible by 9? (a) 5607 (b) 5601 (c) 5614 (d) 5618
8. 8.What is the smallest number you must add to 403 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
9. 9.What is the remainder when 578 is divided by 10? (a) 9 (b) 2 (c) 8 (d) 7
10. 10.Which of these numbers is divisible by 3? (a) 1371 (b) 1369 (c) 1373 (d) 1375
11. 11.Which of these numbers is divisible by 11? (a) 5757 (b) 5752 (c) 5767 (d) 5753
12. 12.What is the smallest number you must add to 581 to make it divisible by 9? (a) 4 (b) 5 (c) 3 (d) 6
13. 13.Which of these numbers is divisible by 3? (a) 128 (b) 121 (c) 123 (d) 125
14. 14.What is the remainder when 141 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 6
15. 15.What is the remainder when 82 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
16. 16.What is the units digit of 3^17? (a) 7 (b) 9 (c) 3 (d) 1
17. 17.What is the smallest number you must add to 429 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
18. 18.Which of these numbers is divisible by 4? (a) 3238 (b) 3241 (c) 3240 (d) 3245
19. 19.What is the remainder when 478 is divided by 9? (a) 0 (b) 8 (c) 2 (d) 1
20. 20.Which of these numbers is divisible by 9? (a) 5584 (b) 5594 (c) 5580 (d) 5572
21. 21.What is the units digit of 4^43? (a) 5 (b) 6 (c) 3 (d) 4
22. 22.What is the smallest number you must add to 601 to make it divisible by 8? (a) 6 (b) 7 (c) 1 (d) 5
23. 23.What is the remainder when 229 is divided by 3? (a) 2 (b) 0 (c) 1
24. 24.Which of these numbers is divisible by 4? (a) 2122 (b) 2120 (c) 2119 (d) 2125
25. 25.What is the remainder when 854 is divided by 12? (a) 10 (b) 2 (c) 3 (d) 1
26. 26.What is the smallest number you must add to 86 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 4
27. 27.Which of these numbers is divisible by 8? (a) 2291 (b) 2274 (c) 2280 (d) 2282
28. 28.Which of these numbers is divisible by 4? (a) 3591 (b) 3587 (c) 3588 (d) 3595
29. 29.What is the smallest number you must add to 357 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
30. 30.What is the remainder when 516 is divided by 9? (a) 4 (b) 2 (c) 6 (d) 3
31. 31.Which of these numbers is divisible by 3? (a) 1246 (b) 1248 (c) 1250 (d) 1252
32. 32.What is the smallest number you must add to 251 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
33. 33.What is the units digit of 4^38? (a) 7 (b) 5 (c) 6 (d) 4
34. 34.Which of these numbers is divisible by 11? (a) 5245 (b) 5225 (c) 5234 (d) 5218
35. 35.What is the units digit of 2^20? (a) 4 (b) 2 (c) 6 (d) 8
36. 36.What is the remainder when 555 is divided by 8? (a) 4 (b) 3 (c) 2 (d) 5
37. 37.What is the remainder when 77 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
38. 38.Which of these numbers is divisible by 6? (a) 1471 (b) 1460 (c) 1465 (d) 1464
39. 39.What is the remainder when 76 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
40. 40.What is the remainder when 151 is divided by 6? (a) 5 (b) 2 (c) 1 (d) 0

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