Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 368 · 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 11? (a) 3382 (b) 3407 (c) 3388 (d) 3393
2. 2.Which of these numbers is divisible by 9? (a) 6558 (b) 6562 (c) 6577 (d) 6561
3. 3.What is the smallest number you must add to 127 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 5
4. 4.What is the units digit of 3^48? (a) 1 (b) 3 (c) 9 (d) 7
5. 5.What is the units digit of 3^39? (a) 3 (b) 1 (c) 7 (d) 9
6. 6.Which of these numbers is divisible by 11? (a) 9577 (b) 9585 (c) 9560 (d) 9570
7. 7.What is the remainder when 541 is divided by 10? (a) 9 (b) 2 (c) 1 (d) 0
8. 8.What is the remainder when 787 is divided by 10? (a) 3 (b) 8 (c) 7 (d) 6
9. 9.What is the units digit of 7^33? (a) 7 (b) 9 (c) 1 (d) 3
10. 10.Which of these numbers is divisible by 11? (a) 4177 (b) 4169 (c) 4167 (d) 4183
11. 11.What is the smallest number you must add to 79 to make it divisible by 3? (a) 1 (b) 2
12. 12.Which of these numbers is divisible by 3? (a) 2275 (b) 2281 (c) 2277 (d) 2278
13. 13.What is the smallest number you must add to 160 to make it divisible by 6? (a) 2 (b) 3 (c) 1 (d) 4
14. 14.What is the units digit of 7^23? (a) 9 (b) 7 (c) 1 (d) 3
15. 15.Which of these numbers is divisible by 3? (a) 558 (b) 562 (c) 560 (d) 556
16. 16.Which of these numbers is divisible by 8? (a) 1040 (b) 1039 (c) 1045 (d) 1053
17. 17.What is the remainder when 653 is divided by 12? (a) 4 (b) 6 (c) 7 (d) 5
18. 18.What is the smallest number you must add to 515 to make it divisible by 11? (a) 2 (b) 1 (c) 3 (d) 9
19. 19.Which of these numbers is divisible by 9? (a) 2879 (b) 2865 (c) 2871 (d) 2884
20. 20.Which of these numbers is divisible by 11? (a) 5517 (b) 5499 (c) 5502 (d) 5500
21. 21.Which of these numbers is divisible by 3? (a) 550 (b) 549 (c) 554 (d) 547
22. 22.Which of these numbers is divisible by 11? (a) 4499 (b) 4493 (c) 4519 (d) 4508
23. 23.What is the remainder when 109 is divided by 3? (a) 1 (b) 0 (c) 2
24. 24.What is the units digit of 8^42? (a) 4 (b) 8 (c) 2 (d) 6
25. 25.Which of these numbers is divisible by 6? (a) 4242 (b) 4244 (c) 4241 (d) 4249
26. 26.What is the remainder when 326 is divided by 9? (a) 2 (b) 3 (c) 1 (d) 7
27. 27.What is the remainder when 241 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
28. 28.What is the smallest number you must add to 238 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
29. 29.What is the remainder when 345 is divided by 11? (a) 4 (b) 5 (c) 3 (d) 7
30. 30.What is the units digit of 7^54? (a) 3 (b) 1 (c) 7 (d) 9
31. 31.What is the units digit of 9^56? (a) 9 (b) 2 (c) 0 (d) 1
32. 32.Which of these numbers is divisible by 6? (a) 4300 (b) 4303 (c) 4309 (d) 4302
33. 33.What is the smallest number you must add to 208 to make it divisible by 7? (a) 3 (b) 1 (c) 2 (d) 5
34. 34.What is the remainder when 63 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
35. 35.Which of these numbers is divisible by 3? (a) 667 (b) 673 (c) 669 (d) 670
36. 36.Which of these numbers is divisible by 4? (a) 2480 (b) 2479 (c) 2482 (d) 2485
37. 37.Which of these numbers is divisible by 4? (a) 2789 (b) 2795 (c) 2799 (d) 2792
38. 38.What is the smallest number you must add to 149 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 1
39. 39.What is the smallest number you must add to 309 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
40. 40.What is the smallest number you must add to 258 to make it divisible by 4? (a) 1 (b) 2 (c) 3

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