Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 672 · 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 9^24? (a) 0 (b) 9 (c) 2 (d) 1
2. 2.Which of these numbers is divisible by 6? (a) 4865 (b) 4860 (c) 4870 (d) 4855
3. 3.What is the smallest number you must add to 341 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 2
4. 4.What is the smallest number you must add to 297 to make it divisible by 7? (a) 3 (b) 4 (c) 6 (d) 5
5. 5.Which of these numbers is divisible by 9? (a) 3267 (b) 3260 (c) 3272 (d) 3282
6. 6.What is the remainder when 254 is divided by 7? (a) 5 (b) 1 (c) 3 (d) 2
7. 7.Which of these numbers is divisible by 3? (a) 322 (b) 328 (c) 324 (d) 325
8. 8.Which of these numbers is divisible by 4? (a) 2482 (b) 2476 (c) 2479 (d) 2473
9. 9.What is the smallest number you must add to 78 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 1
10. 10.What is the units digit of 4^37? (a) 5 (b) 4 (c) 6 (d) 3
11. 11.What is the units digit of 4^54? (a) 4 (b) 7 (c) 6 (d) 5
12. 12.What is the smallest number you must add to 488 to make it divisible by 9? (a) 7 (b) 2 (c) 8 (d) 6
13. 13.Which of these numbers is divisible by 9? (a) 7872 (b) 7866 (c) 7879 (d) 7864
14. 14.What is the units digit of 8^22? (a) 2 (b) 4 (c) 8 (d) 6
15. 15.Which of these numbers is divisible by 11? (a) 2947 (b) 2956 (c) 2960 (d) 2948
16. 16.Which of these numbers is divisible by 4? (a) 3543 (b) 3537 (c) 3540 (d) 3545
17. 17.What is the smallest number you must add to 312 to make it divisible by 7? (a) 3 (b) 5 (c) 2 (d) 4
18. 18.What is the units digit of 3^14? (a) 1 (b) 3 (c) 7 (d) 9
19. 19.Which of these numbers is divisible by 4? (a) 2318 (b) 2322 (c) 2320 (d) 2325
20. 20.What is the units digit of 8^48? (a) 8 (b) 6 (c) 4 (d) 2
21. 21.What is the smallest number you must add to 52 to make it divisible by 3? (a) 2 (b) 1
22. 22.What is the units digit of 4^15? (a) 6 (b) 3 (c) 5 (d) 4
23. 23.What is the units digit of 4^50? (a) 7 (b) 4 (c) 5 (d) 6
24. 24.What is the units digit of 2^50? (a) 8 (b) 4 (c) 2 (d) 6
25. 25.What is the smallest number you must add to 203 to make it divisible by 3? (a) 1 (b) 2
26. 26.What is the remainder when 149 is divided by 3? (a) 2 (b) 0 (c) 1
27. 27.What is the units digit of 8^55? (a) 6 (b) 8 (c) 4 (d) 2
28. 28.Which of these numbers is divisible by 3? (a) 337 (b) 340 (c) 336 (d) 335
29. 29.Which of these numbers is divisible by 4? (a) 714 (b) 710 (c) 707 (d) 708
30. 30.What is the remainder when 96 is divided by 9? (a) 5 (b) 3 (c) 6 (d) 7
31. 31.What is the remainder when 59 is divided by 3? (a) 1 (b) 0 (c) 2
32. 32.What is the smallest number you must add to 623 to make it divisible by 12? (a) 2 (b) 11 (c) 3 (d) 1
33. 33.What is the smallest number you must add to 319 to make it divisible by 4? (a) 1 (b) 2 (c) 3
34. 34.Which of these numbers is divisible by 3? (a) 548 (b) 550 (c) 546 (d) 544
35. 35.What is the units digit of 2^16? (a) 6 (b) 2 (c) 4 (d) 8
36. 36.Which of these numbers is divisible by 3? (a) 1859 (b) 1855 (c) 1852 (d) 1854
37. 37.What is the smallest number you must add to 94 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
38. 38.Which of these numbers is divisible by 8? (a) 536 (b) 531 (c) 538 (d) 549
39. 39.What is the remainder when 451 is divided by 12? (a) 5 (b) 7 (c) 6 (d) 8
40. 40.Which of these numbers is divisible by 3? (a) 2354 (b) 2356 (c) 2359 (d) 2355

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