Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 803 · 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 remainder when 157 is divided by 3? (a) 1 (b) 2 (c) 0
2. 2.What is the remainder when 336 is divided by 11? (a) 6 (b) 5 (c) 7 (d) 8
3. 3.What is the units digit of 2^24? (a) 2 (b) 6 (c) 4 (d) 8
4. 4.What is the smallest number you must add to 31 to make it divisible by 3? (a) 2 (b) 1
5. 5.What is the units digit of 4^12? (a) 7 (b) 6 (c) 4 (d) 5
6. 6.What is the units digit of 4^18? (a) 4 (b) 7 (c) 5 (d) 6
7. 7.What is the units digit of 3^32? (a) 7 (b) 1 (c) 3 (d) 9
8. 8.What is the units digit of 9^56? (a) 2 (b) 0 (c) 9 (d) 1
9. 9.What is the remainder when 558 is divided by 11? (a) 9 (b) 7 (c) 3 (d) 8
10. 10.Which of these numbers is divisible by 9? (a) 5213 (b) 5207 (c) 5225 (d) 5211
11. 11.What is the smallest number you must add to 121 to make it divisible by 12? (a) 9 (b) 1 (c) 11 (d) 10
12. 12.Which of these numbers is divisible by 3? (a) 1166 (b) 1171 (c) 1169 (d) 1167
13. 13.Which of these numbers is divisible by 4? (a) 1401 (b) 1411 (c) 1405 (d) 1404
14. 14.Which of these numbers is divisible by 11? (a) 2946 (b) 2964 (c) 2951 (d) 2948
15. 15.What is the smallest number you must add to 153 to make it divisible by 4? (a) 2 (b) 3 (c) 1
16. 16.What is the remainder when 212 is divided by 10? (a) 2 (b) 1 (c) 8 (d) 3
17. 17.Which of these numbers is divisible by 8? (a) 5712 (b) 5717 (c) 5723 (d) 5706
18. 18.Which of these numbers is divisible by 6? (a) 2199 (b) 2212 (c) 2204 (d) 2202
19. 19.What is the units digit of 9^19? (a) 1 (b) 7 (c) 8 (d) 9
20. 20.What is the units digit of 9^60? (a) 2 (b) 9 (c) 1 (d) 0
21. 21.What is the units digit of 3^56? (a) 1 (b) 7 (c) 3 (d) 9
22. 22.What is the remainder when 233 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
23. 23.What is the smallest number you must add to 136 to make it divisible by 11? (a) 8 (b) 6 (c) 7 (d) 4
24. 24.What is the remainder when 149 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
25. 25.What is the smallest number you must add to 279 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
26. 26.Which of these numbers is divisible by 9? (a) 7275 (b) 7262 (c) 7264 (d) 7263
27. 27.What is the units digit of 2^20? (a) 8 (b) 6 (c) 4 (d) 2
28. 28.What is the units digit of 3^39? (a) 3 (b) 1 (c) 9 (d) 7
29. 29.What is the smallest number you must add to 203 to make it divisible by 8? (a) 3 (b) 5 (c) 4 (d) 6
30. 30.Which of these numbers is divisible by 3? (a) 1585 (b) 1589 (c) 1591 (d) 1587
31. 31.What is the smallest number you must add to 290 to make it divisible by 4? (a) 1 (b) 2 (c) 3
32. 32.What is the remainder when 282 is divided by 10? (a) 8 (b) 1 (c) 3 (d) 2
33. 33.What is the remainder when 57 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
34. 34.What is the units digit of 2^41? (a) 2 (b) 6 (c) 4 (d) 8
35. 35.What is the remainder when 219 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
36. 36.What is the units digit of 4^27? (a) 6 (b) 5 (c) 3 (d) 4
37. 37.What is the remainder when 622 is divided by 10? (a) 3 (b) 8 (c) 1 (d) 2
38. 38.Which of these numbers is divisible by 3? (a) 1376 (b) 1373 (c) 1371 (d) 1370
39. 39.What is the units digit of 2^40? (a) 4 (b) 8 (c) 6 (d) 2
40. 40.What is the remainder when 84 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4

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