Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 293 · 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 350 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
2. 2.Which of these numbers is divisible by 8? (a) 1639 (b) 1617 (c) 1626 (d) 1624
3. 3.Which of these numbers is divisible by 3? (a) 994 (b) 993 (c) 997 (d) 991
4. 4.What is the smallest number you must add to 399 to make it divisible by 11? (a) 9 (b) 7 (c) 8 (d) 3
5. 5.What is the remainder when 243 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
6. 6.What is the units digit of 7^23? (a) 7 (b) 9 (c) 1 (d) 3
7. 7.What is the units digit of 2^14? (a) 2 (b) 8 (c) 6 (d) 4
8. 8.What is the units digit of 9^60? (a) 9 (b) 0 (c) 2 (d) 1
9. 9.What is the units digit of 7^30? (a) 1 (b) 3 (c) 7 (d) 9
10. 10.Which of these numbers is divisible by 9? (a) 768 (b) 761 (c) 756 (d) 751
11. 11.What is the remainder when 68 is divided by 3? (a) 1 (b) 2 (c) 0
12. 12.What is the units digit of 7^34? (a) 7 (b) 3 (c) 1 (d) 9
13. 13.What is the units digit of 4^52? (a) 4 (b) 6 (c) 7 (d) 5
14. 14.What is the smallest number you must add to 557 to make it divisible by 7? (a) 2 (b) 5 (c) 4 (d) 3
15. 15.What is the smallest number you must add to 121 to make it divisible by 4? (a) 1 (b) 3 (c) 2
16. 16.What is the smallest number you must add to 623 to make it divisible by 8? (a) 2 (b) 3 (c) 7 (d) 1
17. 17.What is the smallest number you must add to 142 to make it divisible by 8? (a) 6 (b) 3 (c) 1 (d) 2
18. 18.What is the units digit of 4^25? (a) 3 (b) 6 (c) 5 (d) 4
19. 19.What is the remainder when 467 is divided by 7? (a) 5 (b) 4 (c) 2 (d) 6
20. 20.What is the remainder when 110 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
21. 21.What is the units digit of 9^14? (a) 0 (b) 9 (c) 1 (d) 2
22. 22.What is the remainder when 147 is divided by 10? (a) 7 (b) 8 (c) 3 (d) 6
23. 23.Which of these numbers is divisible by 3? (a) 1278 (b) 1280 (c) 1276 (d) 1282
24. 24.What is the smallest number you must add to 219 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
25. 25.What is the units digit of 3^60? (a) 7 (b) 9 (c) 1 (d) 3
26. 26.What is the remainder when 132 is divided by 9? (a) 6 (b) 5 (c) 3 (d) 7
27. 27.Which of these numbers is divisible by 3? (a) 2230 (b) 2236 (c) 2234 (d) 2232
28. 28.What is the remainder when 301 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
29. 29.Which of these numbers is divisible by 6? (a) 3265 (b) 3258 (c) 3255 (d) 3261
30. 30.What is the smallest number you must add to 778 to make it divisible by 12? (a) 10 (b) 3 (c) 2 (d) 1
31. 31.What is the smallest number you must add to 100 to make it divisible by 3? (a) 2 (b) 1
32. 32.What is the units digit of 8^19? (a) 4 (b) 8 (c) 2 (d) 6
33. 33.What is the units digit of 3^23? (a) 1 (b) 9 (c) 7 (d) 3
34. 34.Which of these numbers is divisible by 3? (a) 2411 (b) 2406 (c) 2405 (d) 2408
35. 35.What is the remainder when 353 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
36. 36.What is the remainder when 380 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
37. 37.What is the smallest number you must add to 114 to make it divisible by 8? (a) 5 (b) 7 (c) 2 (d) 6
38. 38.What is the smallest number you must add to 155 to make it divisible by 3? (a) 2 (b) 1
39. 39.What is the remainder when 100 is divided by 3? (a) 0 (b) 1 (c) 2
40. 40.What is the remainder when 509 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 2

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