Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 769 · 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 93 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 2
2. 2.Which of these numbers is divisible by 9? (a) 2452 (b) 2448 (c) 2444 (d) 2463
3. 3.What is the remainder when 186 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
4. 4.What is the remainder when 376 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
5. 5.What is the remainder when 113 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
6. 6.Which of these numbers is divisible by 11? (a) 2229 (b) 2240 (c) 2220 (d) 2222
7. 7.What is the remainder when 356 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
8. 8.What is the units digit of 9^24? (a) 0 (b) 2 (c) 1 (d) 9
9. 9.What is the units digit of 3^20? (a) 9 (b) 7 (c) 3 (d) 1
10. 10.What is the smallest number you must add to 109 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 4
11. 11.What is the units digit of 2^58? (a) 8 (b) 2 (c) 4 (d) 6
12. 12.Which of these numbers is divisible by 8? (a) 1953 (b) 1960 (c) 1966 (d) 1970
13. 13.What is the remainder when 249 is divided by 8? (a) 0 (b) 1 (c) 7 (d) 2
14. 14.What is the remainder when 285 is divided by 10? (a) 6 (b) 7 (c) 5 (d) 4
15. 15.What is the remainder when 43 is divided by 3? (a) 0 (b) 2 (c) 1
16. 16.What is the smallest number you must add to 125 to make it divisible by 4? (a) 2 (b) 1 (c) 3
17. 17.What is the smallest number you must add to 130 to make it divisible by 8? (a) 6 (b) 5 (c) 7 (d) 2
18. 18.What is the remainder when 175 is divided by 6? (a) 0 (b) 2 (c) 5 (d) 1
19. 19.Which of these numbers is divisible by 3? (a) 1819 (b) 1823 (c) 1818 (d) 1817
20. 20.What is the smallest number you must add to 110 to make it divisible by 4? (a) 1 (b) 2 (c) 3
21. 21.What is the smallest number you must add to 705 to make it divisible by 10? (a) 4 (b) 6 (c) 5 (d) 7
22. 22.Which of these numbers is divisible by 4? (a) 2491 (b) 2495 (c) 2497 (d) 2492
23. 23.Which of these numbers is divisible by 6? (a) 1458 (b) 1468 (c) 1454 (d) 1460
24. 24.What is the units digit of 3^57? (a) 7 (b) 9 (c) 3 (d) 1
25. 25.What is the units digit of 8^26? (a) 6 (b) 2 (c) 4 (d) 8
26. 26.What is the units digit of 8^22? (a) 2 (b) 8 (c) 6 (d) 4
27. 27.What is the remainder when 223 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
28. 28.What is the remainder when 246 is divided by 7? (a) 6 (b) 0 (c) 2 (d) 1
29. 29.What is the smallest number you must add to 485 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 5
30. 30.What is the units digit of 3^11? (a) 1 (b) 9 (c) 7 (d) 3
31. 31.Which of these numbers is divisible by 4? (a) 2351 (b) 2346 (c) 2348 (d) 2354
32. 32.Which of these numbers is divisible by 3? (a) 1126 (b) 1124 (c) 1120 (d) 1122
33. 33.Which of these numbers is divisible by 6? (a) 2736 (b) 2743 (c) 2739 (d) 2732
34. 34.What is the units digit of 7^32? (a) 1 (b) 7 (c) 9 (d) 3
35. 35.What is the units digit of 9^14? (a) 9 (b) 0 (c) 1 (d) 2
36. 36.What is the units digit of 2^14? (a) 2 (b) 4 (c) 6 (d) 8
37. 37.What is the units digit of 7^52? (a) 7 (b) 9 (c) 3 (d) 1
38. 38.What is the remainder when 194 is divided by 7? (a) 6 (b) 5 (c) 2 (d) 4
39. 39.Which of these numbers is divisible by 9? (a) 1135 (b) 1133 (c) 1120 (d) 1125
40. 40.Which of these numbers is divisible by 3? (a) 1240 (b) 1242 (c) 1244 (d) 1246

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