Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 493 · 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 smallest number you must add to 97 to make it divisible by 4? (a) 2 (b) 3 (c) 1
2. 2.What is the smallest number you must add to 178 to make it divisible by 8? (a) 2 (b) 7 (c) 6 (d) 5
3. 3.What is the remainder when 137 is divided by 10? (a) 8 (b) 6 (c) 7 (d) 3
4. 4.What is the units digit of 3^41? (a) 1 (b) 9 (c) 3 (d) 7
5. 5.Which of these numbers is divisible by 8? (a) 5234 (b) 5226 (c) 5247 (d) 5232
6. 6.What is the units digit of 3^12? (a) 7 (b) 3 (c) 9 (d) 1
7. 7.What is the units digit of 9^41? (a) 9 (b) 8 (c) 1 (d) 7
8. 8.Which of these numbers is divisible by 9? (a) 3674 (b) 3685 (c) 3672 (d) 3668
9. 9.Which of these numbers is divisible by 3? (a) 220 (b) 214 (c) 218 (d) 216
10. 10.Which of these numbers is divisible by 3? (a) 1360 (b) 1356 (c) 1358 (d) 1355
11. 11.Which of these numbers is divisible by 11? (a) 7788 (b) 7787 (c) 7803 (d) 7793
12. 12.What is the smallest number you must add to 151 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the smallest number you must add to 229 to make it divisible by 11? (a) 9 (b) 2 (c) 3 (d) 1
14. 14.What is the remainder when 311 is divided by 8? (a) 7 (b) 1 (c) 6 (d) 5
15. 15.What is the remainder when 166 is divided by 9? (a) 6 (b) 5 (c) 3 (d) 4
16. 16.What is the smallest number you must add to 101 to make it divisible by 3? (a) 1 (b) 2
17. 17.What is the units digit of 8^59? (a) 8 (b) 6 (c) 4 (d) 2
18. 18.What is the units digit of 7^47? (a) 3 (b) 9 (c) 1 (d) 7
19. 19.What is the remainder when 646 is divided by 12? (a) 2 (b) 9 (c) 11 (d) 10
20. 20.What is the remainder when 220 is divided by 3? (a) 0 (b) 1 (c) 2
21. 21.Which of these numbers is divisible by 4? (a) 3326 (b) 3320 (c) 3322 (d) 3319
22. 22.What is the units digit of 8^26? (a) 6 (b) 2 (c) 8 (d) 4
23. 23.What is the smallest number you must add to 154 to make it divisible by 4? (a) 2 (b) 1 (c) 3
24. 24.What is the remainder when 554 is divided by 10? (a) 5 (b) 3 (c) 6 (d) 4
25. 25.What is the units digit of 2^11? (a) 8 (b) 6 (c) 2 (d) 4
26. 26.What is the smallest number you must add to 174 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 6
27. 27.Which of these numbers is divisible by 11? (a) 5955 (b) 5968 (c) 5951 (d) 5944
28. 28.Which of these numbers is divisible by 6? (a) 5247 (b) 5242 (c) 5244 (d) 5253
29. 29.What is the remainder when 855 is divided by 11? (a) 3 (b) 8 (c) 7 (d) 9
30. 30.What is the remainder when 125 is divided by 3? (a) 2 (b) 0 (c) 1
31. 31.Which of these numbers is divisible by 8? (a) 778 (b) 786 (c) 784 (d) 794
32. 32.Which of these numbers is divisible by 4? (a) 2450 (b) 2452 (c) 2457 (d) 2455
33. 33.What is the units digit of 8^14? (a) 2 (b) 8 (c) 4 (d) 6
34. 34.What is the remainder when 600 is divided by 9? (a) 5 (b) 3 (c) 7 (d) 6
35. 35.What is the remainder when 496 is divided by 11? (a) 1 (b) 2 (c) 0 (d) 10
36. 36.Which of these numbers is divisible by 4? (a) 339 (b) 343 (c) 336 (d) 335
37. 37.Which of these numbers is divisible by 11? (a) 6842 (b) 6835 (c) 6861 (d) 6845
38. 38.What is the smallest number you must add to 301 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
39. 39.What is the remainder when 230 is divided by 6? (a) 1 (b) 4 (c) 2 (d) 3
40. 40.What is the units digit of 3^14? (a) 1 (b) 3 (c) 9 (d) 7

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