Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 484 · 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) 3288 (b) 3296 (c) 3291 (d) 3283
2. 2.What is the smallest number you must add to 169 to make it divisible by 11? (a) 4 (b) 7 (c) 6 (d) 8
3. 3.What is the units digit of 9^51? (a) 1 (b) 8 (c) 9 (d) 7
4. 4.Which of these numbers is divisible by 3? (a) 436 (b) 431 (c) 432 (d) 434
5. 5.What is the smallest number you must add to 115 to make it divisible by 4? (a) 2 (b) 1 (c) 3
6. 6.What is the units digit of 7^21? (a) 3 (b) 9 (c) 7 (d) 1
7. 7.Which of these numbers is divisible by 8? (a) 3320 (b) 3331 (c) 3316 (d) 3323
8. 8.What is the smallest number you must add to 97 to make it divisible by 3? (a) 1 (b) 2
9. 9.What is the remainder when 314 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
10. 10.What is the units digit of 2^16? (a) 8 (b) 2 (c) 4 (d) 6
11. 11.What is the units digit of 4^58? (a) 4 (b) 6 (c) 5 (d) 7
12. 12.What is the remainder when 799 is divided by 10? (a) 1 (b) 8 (c) 7 (d) 9
13. 13.What is the smallest number you must add to 149 to make it divisible by 3? (a) 1 (b) 2
14. 14.Which of these numbers is divisible by 8? (a) 881 (b) 872 (c) 871 (d) 876
15. 15.What is the units digit of 2^46? (a) 2 (b) 6 (c) 4 (d) 8
16. 16.Which of these numbers is divisible by 6? (a) 788 (b) 802 (c) 795 (d) 792
17. 17.What is the units digit of 8^58? (a) 8 (b) 4 (c) 2 (d) 6
18. 18.What is the units digit of 3^53? (a) 9 (b) 7 (c) 1 (d) 3
19. 19.What is the smallest number you must add to 418 to make it divisible by 10? (a) 1 (b) 3 (c) 8 (d) 2
20. 20.What is the units digit of 2^20? (a) 4 (b) 2 (c) 8 (d) 6
21. 21.What is the remainder when 115 is divided by 3? (a) 0 (b) 2 (c) 1
22. 22.What is the remainder when 805 is divided by 10? (a) 5 (b) 7 (c) 4 (d) 6
23. 23.What is the remainder when 114 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
24. 24.What is the units digit of 4^32? (a) 5 (b) 6 (c) 4 (d) 7
25. 25.Which of these numbers is divisible by 8? (a) 3406 (b) 3389 (c) 3392 (d) 3397
26. 26.What is the smallest number you must add to 107 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 6
27. 27.What is the units digit of 7^56? (a) 3 (b) 1 (c) 7 (d) 9
28. 28.What is the units digit of 8^52? (a) 4 (b) 2 (c) 6 (d) 8
29. 29.What is the units digit of 4^48? (a) 6 (b) 5 (c) 7 (d) 4
30. 30.Which of these numbers is divisible by 9? (a) 4407 (b) 4393 (c) 4401 (d) 4417
31. 31.What is the units digit of 7^40? (a) 1 (b) 3 (c) 7 (d) 9
32. 32.What is the units digit of 3^26? (a) 7 (b) 3 (c) 1 (d) 9
33. 33.What is the smallest number you must add to 145 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 1
34. 34.What is the smallest number you must add to 309 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 1
35. 35.What is the remainder when 90 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
36. 36.Which of these numbers is divisible by 8? (a) 5559 (b) 5572 (c) 5560 (d) 5562
37. 37.What is the units digit of 7^57? (a) 3 (b) 9 (c) 7 (d) 1
38. 38.Which of these numbers is divisible by 8? (a) 3305 (b) 3312 (c) 3323 (d) 3315
39. 39.What is the remainder when 426 is divided by 7? (a) 6 (b) 4 (c) 1 (d) 5
40. 40.What is the units digit of 8^33? (a) 2 (b) 6 (c) 4 (d) 8

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