Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 257 · 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 4^30? (a) 5 (b) 6 (c) 7 (d) 4
2. 2.What is the remainder when 124 is divided by 3? (a) 1 (b) 0 (c) 2
3. 3.What is the remainder when 355 is divided by 7? (a) 5 (b) 4 (c) 2 (d) 6
4. 4.What is the remainder when 185 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
5. 5.What is the units digit of 9^34? (a) 2 (b) 0 (c) 9 (d) 1
6. 6.What is the smallest number you must add to 172 to make it divisible by 10? (a) 9 (b) 2 (c) 7 (d) 8
7. 7.Which of these numbers is divisible by 3? (a) 1375 (b) 1369 (c) 1373 (d) 1371
8. 8.What is the units digit of 8^40? (a) 8 (b) 6 (c) 4 (d) 2
9. 9.What is the units digit of 4^44? (a) 4 (b) 5 (c) 7 (d) 6
10. 10.What is the units digit of 4^51? (a) 4 (b) 3 (c) 5 (d) 6
11. 11.What is the smallest number you must add to 218 to make it divisible by 9? (a) 2 (b) 7 (c) 8 (d) 6
12. 12.What is the smallest number you must add to 151 to make it divisible by 7? (a) 4 (b) 5 (c) 2 (d) 3
13. 13.What is the remainder when 403 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
14. 14.What is the smallest number you must add to 315 to make it divisible by 12? (a) 10 (b) 8 (c) 9 (d) 3
15. 15.What is the remainder when 537 is divided by 8? (a) 0 (b) 2 (c) 7 (d) 1
16. 16.What is the units digit of 8^31? (a) 4 (b) 8 (c) 6 (d) 2
17. 17.What is the remainder when 227 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
18. 18.Which of these numbers is divisible by 4? (a) 1791 (b) 1792 (c) 1798 (d) 1794
19. 19.What is the units digit of 7^36? (a) 1 (b) 3 (c) 7 (d) 9
20. 20.What is the units digit of 4^22? (a) 6 (b) 5 (c) 4 (d) 7
21. 21.Which of these numbers is divisible by 11? (a) 2570 (b) 2579 (c) 2555 (d) 2563
22. 22.Which of these numbers is divisible by 3? (a) 506 (b) 504 (c) 502 (d) 509
23. 23.What is the units digit of 7^39? (a) 1 (b) 3 (c) 9 (d) 7
24. 24.What is the remainder when 309 is divided by 9? (a) 2 (b) 4 (c) 3 (d) 6
25. 25.Which of these numbers is divisible by 4? (a) 764 (b) 763 (c) 766 (d) 771
26. 26.Which of these numbers is divisible by 8? (a) 2276 (b) 2266 (c) 2284 (d) 2272
27. 27.What is the smallest number you must add to 577 to make it divisible by 11? (a) 7 (b) 6 (c) 8 (d) 5
28. 28.What is the units digit of 7^44? (a) 7 (b) 1 (c) 9 (d) 3
29. 29.What is the remainder when 176 is divided by 10? (a) 7 (b) 6 (c) 4 (d) 5
30. 30.What is the units digit of 3^31? (a) 3 (b) 1 (c) 9 (d) 7
31. 31.What is the smallest number you must add to 166 to make it divisible by 9? (a) 6 (b) 5 (c) 7 (d) 4
32. 32.Which of these numbers is divisible by 4? (a) 2794 (b) 2803 (c) 2796 (d) 2799
33. 33.What is the smallest number you must add to 225 to make it divisible by 8? (a) 7 (b) 1 (c) 6 (d) 5
34. 34.What is the units digit of 8^12? (a) 4 (b) 2 (c) 6 (d) 8
35. 35.Which of these numbers is divisible by 6? (a) 2425 (b) 2416 (c) 2418 (d) 2423
36. 36.Which of these numbers is divisible by 4? (a) 1601 (b) 1607 (c) 1610 (d) 1604
37. 37.What is the smallest number you must add to 75 to make it divisible by 4? (a) 3 (b) 1 (c) 2
38. 38.What is the smallest number you must add to 182 to make it divisible by 3? (a) 1 (b) 2
39. 39.What is the units digit of 2^42? (a) 8 (b) 2 (c) 6 (d) 4
40. 40.What is the units digit of 3^40? (a) 1 (b) 3 (c) 9 (d) 7

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