Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 654 · 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 243 is divided by 10? (a) 4 (b) 3 (c) 2 (d) 7
2. 2.What is the units digit of 9^13? (a) 8 (b) 9 (c) 1 (d) 7
3. 3.What is the smallest number you must add to 327 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
4. 4.What is the smallest number you must add to 784 to make it divisible by 12? (a) 9 (b) 7 (c) 4 (d) 8
5. 5.What is the units digit of 7^52? (a) 7 (b) 3 (c) 1 (d) 9
6. 6.What is the remainder when 351 is divided by 7? (a) 0 (b) 6 (c) 2 (d) 1
7. 7.Which of these numbers is divisible by 4? (a) 1991 (b) 1987 (c) 1993 (d) 1988
8. 8.Which of these numbers is divisible by 3? (a) 2035 (b) 2034 (c) 2039 (d) 2033
9. 9.Which of these numbers is divisible by 4? (a) 1155 (b) 1161 (c) 1158 (d) 1156
10. 10.Which of these numbers is divisible by 9? (a) 4496 (b) 4502 (c) 4486 (d) 4491
11. 11.Which of these numbers is divisible by 6? (a) 4657 (b) 4669 (c) 4664 (d) 4662
12. 12.What is the units digit of 4^39? (a) 6 (b) 5 (c) 4 (d) 3
13. 13.What is the smallest number you must add to 153 to make it divisible by 4? (a) 3 (b) 1 (c) 2
14. 14.What is the smallest number you must add to 598 to make it divisible by 11? (a) 4 (b) 8 (c) 6 (d) 7
15. 15.What is the remainder when 664 is divided by 12? (a) 5 (b) 3 (c) 8 (d) 4
16. 16.What is the units digit of 3^50? (a) 7 (b) 3 (c) 1 (d) 9
17. 17.What is the remainder when 86 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
18. 18.What is the remainder when 254 is divided by 9? (a) 3 (b) 1 (c) 7 (d) 2
19. 19.What is the units digit of 9^59? (a) 9 (b) 8 (c) 7 (d) 1
20. 20.What is the units digit of 9^42? (a) 9 (b) 1 (c) 0 (d) 2
21. 21.What is the units digit of 2^24? (a) 4 (b) 2 (c) 6 (d) 8
22. 22.What is the remainder when 151 is divided by 3? (a) 0 (b) 1 (c) 2
23. 23.What is the remainder when 365 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 5
24. 24.What is the remainder when 55 is divided by 3? (a) 0 (b) 2 (c) 1
25. 25.What is the remainder when 161 is divided by 3? (a) 1 (b) 2 (c) 0
26. 26.What is the units digit of 7^35? (a) 9 (b) 7 (c) 3 (d) 1
27. 27.What is the smallest number you must add to 69 to make it divisible by 4? (a) 2 (b) 3 (c) 1
28. 28.What is the remainder when 461 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 5
29. 29.What is the remainder when 182 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
30. 30.Which of these numbers is divisible by 4? (a) 2690 (b) 2699 (c) 2692 (d) 2694
31. 31.Which of these numbers is divisible by 11? (a) 1199 (b) 1213 (c) 1197 (d) 1203
32. 32.What is the remainder when 502 is divided by 12? (a) 9 (b) 11 (c) 2 (d) 10
33. 33.What is the remainder when 241 is divided by 6? (a) 5 (b) 1 (c) 2 (d) 0
34. 34.What is the remainder when 181 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
35. 35.What is the units digit of 7^22? (a) 1 (b) 7 (c) 3 (d) 9
36. 36.What is the remainder when 304 is divided by 9? (a) 8 (b) 7 (c) 2 (d) 6
37. 37.Which of these numbers is divisible by 3? (a) 616 (b) 610 (c) 612 (d) 614
38. 38.What is the remainder when 238 is divided by 3? (a) 1 (b) 0 (c) 2
39. 39.Which of these numbers is divisible by 9? (a) 403 (b) 416 (c) 407 (d) 405
40. 40.What is the units digit of 2^14? (a) 6 (b) 8 (c) 2 (d) 4

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