Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 831 · 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 9^22? (a) 0 (b) 9 (c) 1 (d) 2
2. 2.What is the units digit of 7^31? (a) 1 (b) 7 (c) 3 (d) 9
3. 3.What is the smallest number you must add to 115 to make it divisible by 7? (a) 5 (b) 3 (c) 6 (d) 4
4. 4.What is the smallest number you must add to 323 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
5. 5.Which of these numbers is divisible by 4? (a) 2627 (b) 2635 (c) 2628 (d) 2629
6. 6.Which of these numbers is divisible by 3? (a) 1198 (b) 1195 (c) 1194 (d) 1193
7. 7.Which of these numbers is divisible by 8? (a) 538 (b) 536 (c) 546 (d) 535
8. 8.What is the units digit of 2^34? (a) 2 (b) 4 (c) 8 (d) 6
9. 9.What is the remainder when 65 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
10. 10.What is the units digit of 2^60? (a) 2 (b) 6 (c) 8 (d) 4
11. 11.Which of these numbers is divisible by 4? (a) 2544 (b) 2542 (c) 2549 (d) 2547
12. 12.What is the remainder when 246 is divided by 7? (a) 6 (b) 2 (c) 1 (d) 0
13. 13.Which of these numbers is divisible by 11? (a) 8119 (b) 8099 (c) 8109 (d) 8107
14. 14.What is the remainder when 674 is divided by 11? (a) 4 (b) 2 (c) 3 (d) 8
15. 15.What is the units digit of 4^47? (a) 5 (b) 3 (c) 4 (d) 6
16. 16.What is the units digit of 9^56? (a) 2 (b) 0 (c) 9 (d) 1
17. 17.Which of these numbers is divisible by 8? (a) 2947 (b) 2957 (c) 2944 (d) 2939
18. 18.Which of these numbers is divisible by 8? (a) 5128 (b) 5140 (c) 5121 (d) 5132
19. 19.What is the smallest number you must add to 101 to make it divisible by 3? (a) 2 (b) 1
20. 20.What is the remainder when 132 is divided by 10? (a) 3 (b) 2 (c) 8 (d) 1
21. 21.What is the units digit of 4^39? (a) 6 (b) 4 (c) 5 (d) 3
22. 22.What is the remainder when 515 is divided by 12? (a) 1 (b) 9 (c) 10 (d) 11
23. 23.What is the smallest number you must add to 193 to make it divisible by 4? (a) 1 (b) 3 (c) 2
24. 24.What is the smallest number you must add to 693 to make it divisible by 10? (a) 6 (b) 8 (c) 3 (d) 7
25. 25.What is the smallest number you must add to 81 to make it divisible by 8? (a) 5 (b) 1 (c) 7 (d) 6
26. 26.What is the units digit of 3^13? (a) 1 (b) 3 (c) 9 (d) 7
27. 27.Which of these numbers is divisible by 4? (a) 2307 (b) 2299 (c) 2302 (d) 2300
28. 28.What is the units digit of 8^37? (a) 4 (b) 2 (c) 8 (d) 6
29. 29.Which of these numbers is divisible by 3? (a) 1202 (b) 1204 (c) 1199 (d) 1200
30. 30.Which of these numbers is divisible by 3? (a) 2231 (b) 2234 (c) 2228 (d) 2229
31. 31.Which of these numbers is divisible by 6? (a) 2315 (b) 2319 (c) 2305 (d) 2310
32. 32.What is the units digit of 7^16? (a) 7 (b) 3 (c) 9 (d) 1
33. 33.Which of these numbers is divisible by 3? (a) 469 (b) 476 (c) 472 (d) 471
34. 34.What is the smallest number you must add to 373 to make it divisible by 7? (a) 4 (b) 5 (c) 2 (d) 6
35. 35.What is the smallest number you must add to 305 to make it divisible by 8? (a) 7 (b) 6 (c) 1 (d) 5
36. 36.Which of these numbers is divisible by 6? (a) 883 (b) 882 (c) 890 (d) 880
37. 37.Which of these numbers is divisible by 6? (a) 2049 (b) 2039 (c) 2043 (d) 2040
38. 38.What is the smallest number you must add to 874 to make it divisible by 11? (a) 6 (b) 5 (c) 7 (d) 8
39. 39.What is the remainder when 188 is divided by 3? (a) 1 (b) 0 (c) 2
40. 40.What is the smallest number you must add to 234 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4

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