Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 17 · 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 6? (a) 1040 (b) 1054 (c) 1044 (d) 1049
2. 2.What is the smallest number you must add to 855 to make it divisible by 12? (a) 3 (b) 9 (c) 8 (d) 10
3. 3.What is the remainder when 183 is divided by 8? (a) 7 (b) 5 (c) 1 (d) 6
4. 4.What is the smallest number you must add to 271 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 4
5. 5.What is the smallest number you must add to 352 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
6. 6.Which of these numbers is divisible by 9? (a) 5224 (b) 5203 (c) 5213 (d) 5211
7. 7.Which of these numbers is divisible by 11? (a) 4564 (b) 4553 (c) 4554 (d) 4567
8. 8.What is the units digit of 2^44? (a) 4 (b) 2 (c) 6 (d) 8
9. 9.What is the smallest number you must add to 532 to make it divisible by 12? (a) 9 (b) 8 (c) 4 (d) 7
10. 10.What is the remainder when 491 is divided by 7? (a) 1 (b) 0 (c) 2 (d) 6
11. 11.What is the units digit of 9^26? (a) 9 (b) 2 (c) 1 (d) 0
12. 12.What is the smallest number you must add to 152 to make it divisible by 10? (a) 8 (b) 2 (c) 9 (d) 7
13. 13.What is the units digit of 7^47? (a) 7 (b) 9 (c) 1 (d) 3
14. 14.What is the units digit of 9^15? (a) 7 (b) 9 (c) 8 (d) 1
15. 15.Which of these numbers is divisible by 9? (a) 6966 (b) 6981 (c) 6973 (d) 6965
16. 16.Which of these numbers is divisible by 6? (a) 2999 (b) 3001 (c) 2994 (d) 2992
17. 17.What is the smallest number you must add to 445 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 5
18. 18.Which of these numbers is divisible by 9? (a) 3945 (b) 3942 (c) 3953 (d) 3935
19. 19.What is the units digit of 8^44? (a) 2 (b) 8 (c) 4 (d) 6
20. 20.What is the units digit of 9^38? (a) 1 (b) 0 (c) 9 (d) 2
21. 21.What is the remainder when 238 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 2
22. 22.What is the smallest number you must add to 194 to make it divisible by 11? (a) 4 (b) 7 (c) 5 (d) 3
23. 23.Which of these numbers is divisible by 3? (a) 1438 (b) 1437 (c) 1442 (d) 1436
24. 24.What is the smallest number you must add to 197 to make it divisible by 3? (a) 1 (b) 2
25. 25.What is the units digit of 2^47? (a) 4 (b) 6 (c) 2 (d) 8
26. 26.What is the smallest number you must add to 214 to make it divisible by 10? (a) 5 (b) 4 (c) 7 (d) 6
27. 27.What is the smallest number you must add to 373 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
28. 28.What is the units digit of 8^36? (a) 6 (b) 2 (c) 8 (d) 4
29. 29.Which of these numbers is divisible by 6? (a) 900 (b) 899 (c) 901 (d) 909
30. 30.What is the smallest number you must add to 329 to make it divisible by 9? (a) 3 (b) 5 (c) 4 (d) 6
31. 31.What is the units digit of 3^23? (a) 9 (b) 3 (c) 7 (d) 1
32. 32.What is the units digit of 4^56? (a) 7 (b) 6 (c) 5 (d) 4
33. 33.What is the units digit of 7^28? (a) 9 (b) 3 (c) 1 (d) 7
34. 34.What is the units digit of 8^23? (a) 4 (b) 8 (c) 6 (d) 2
35. 35.What is the units digit of 2^40? (a) 8 (b) 4 (c) 6 (d) 2
36. 36.What is the smallest number you must add to 276 to make it divisible by 9? (a) 3 (b) 4 (c) 2 (d) 6
37. 37.Which of these numbers is divisible by 8? (a) 6772 (b) 6760 (c) 6759 (d) 6767
38. 38.What is the remainder when 259 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
39. 39.Which of these numbers is divisible by 4? (a) 190 (b) 193 (c) 192 (d) 199
40. 40.What is the smallest number you must add to 134 to make it divisible by 3? (a) 1 (b) 2

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