Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 991 · 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.Which of these numbers is divisible by 3? (a) 1440 (b) 1444 (c) 1438 (d) 1442
2. 2.What is the remainder when 637 is divided by 10? (a) 8 (b) 7 (c) 3 (d) 6
3. 3.Which of these numbers is divisible by 9? (a) 1310 (b) 1317 (c) 1314 (d) 1324
4. 4.What is the units digit of 2^49? (a) 6 (b) 4 (c) 2 (d) 8
5. 5.Which of these numbers is divisible by 8? (a) 7136 (b) 7146 (c) 7139 (d) 7130
6. 6.What is the smallest number you must add to 299 to make it divisible by 7? (a) 1 (b) 5 (c) 3 (d) 2
7. 7.What is the smallest number you must add to 261 to make it divisible by 4? (a) 3 (b) 2 (c) 1
8. 8.What is the units digit of 3^48? (a) 7 (b) 9 (c) 3 (d) 1
9. 9.Which of these numbers is divisible by 6? (a) 535 (b) 543 (c) 548 (d) 540
10. 10.What is the remainder when 69 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
11. 11.Which of these numbers is divisible by 3? (a) 331 (b) 330 (c) 328 (d) 334
12. 12.Which of these numbers is divisible by 11? (a) 8713 (b) 8709 (c) 8701 (d) 8700
13. 13.What is the units digit of 8^22? (a) 2 (b) 8 (c) 6 (d) 4
14. 14.What is the remainder when 680 is divided by 11? (a) 9 (b) 8 (c) 10 (d) 2
15. 15.What is the units digit of 8^18? (a) 8 (b) 6 (c) 2 (d) 4
16. 16.What is the smallest number you must add to 284 to make it divisible by 10? (a) 5 (b) 7 (c) 4 (d) 6
17. 17.Which of these numbers is divisible by 9? (a) 3143 (b) 3154 (c) 3141 (d) 3139
18. 18.What is the remainder when 151 is divided by 8? (a) 6 (b) 5 (c) 1 (d) 7
19. 19.What is the units digit of 3^34? (a) 3 (b) 1 (c) 7 (d) 9
20. 20.What is the smallest number you must add to 416 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 2
21. 21.What is the remainder when 223 is divided by 7? (a) 5 (b) 4 (c) 1 (d) 6
22. 22.Which of these numbers is divisible by 4? (a) 817 (b) 822 (c) 820 (d) 825
23. 23.What is the remainder when 219 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
24. 24.What is the smallest number you must add to 168 to make it divisible by 10? (a) 8 (b) 1 (c) 2 (d) 3
25. 25.What is the units digit of 9^17? (a) 7 (b) 1 (c) 8 (d) 9
26. 26.What is the smallest number you must add to 130 to make it divisible by 3? (a) 1 (b) 2
27. 27.What is the remainder when 505 is divided by 10? (a) 5 (b) 6 (c) 7 (d) 4
28. 28.Which of these numbers is divisible by 3? (a) 1510 (b) 1517 (c) 1512 (d) 1513
29. 29.What is the smallest number you must add to 219 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
30. 30.What is the units digit of 4^55? (a) 3 (b) 4 (c) 5 (d) 6
31. 31.Which of these numbers is divisible by 8? (a) 2304 (b) 2307 (c) 2319 (d) 2303
32. 32.What is the remainder when 151 is divided by 10? (a) 9 (b) 0 (c) 2 (d) 1
33. 33.What is the smallest number you must add to 177 to make it divisible by 10? (a) 2 (b) 4 (c) 3 (d) 7
34. 34.What is the smallest number you must add to 711 to make it divisible by 12? (a) 10 (b) 8 (c) 3 (d) 9
35. 35.Which of these numbers is divisible by 9? (a) 5059 (b) 5056 (c) 5070 (d) 5058
36. 36.What is the units digit of 7^30? (a) 9 (b) 7 (c) 3 (d) 1
37. 37.Which of these numbers is divisible by 3? (a) 2108 (b) 2111 (c) 2105 (d) 2106
38. 38.Which of these numbers is divisible by 3? (a) 2278 (b) 2273 (c) 2276 (d) 2274
39. 39.Which of these numbers is divisible by 9? (a) 2005 (b) 1998 (c) 1991 (d) 2015
40. 40.What is the smallest number you must add to 70 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 1

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