Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 792 · 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 7^20? (a) 1 (b) 7 (c) 3 (d) 9
2. 2.Which of these numbers is divisible by 6? (a) 239 (b) 240 (c) 250 (d) 244
3. 3.Which of these numbers is divisible by 9? (a) 1278 (b) 1272 (c) 1281 (d) 1289
4. 4.What is the units digit of 7^60? (a) 3 (b) 9 (c) 1 (d) 7
5. 5.What is the units digit of 4^42? (a) 4 (b) 6 (c) 5 (d) 7
6. 6.What is the remainder when 384 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
7. 7.What is the smallest number you must add to 254 to make it divisible by 4? (a) 3 (b) 1 (c) 2
8. 8.What is the smallest number you must add to 435 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
9. 9.What is the smallest number you must add to 133 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 5
10. 10.What is the smallest number you must add to 117 to make it divisible by 7? (a) 5 (b) 3 (c) 2 (d) 1
11. 11.What is the units digit of 2^49? (a) 6 (b) 8 (c) 2 (d) 4
12. 12.Which of these numbers is divisible by 9? (a) 1476 (b) 1482 (c) 1487 (d) 1473
13. 13.Which of these numbers is divisible by 4? (a) 2955 (b) 2956 (c) 2963 (d) 2957
14. 14.What is the remainder when 82 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
15. 15.What is the units digit of 7^48? (a) 3 (b) 7 (c) 9 (d) 1
16. 16.What is the units digit of 4^32? (a) 5 (b) 6 (c) 7 (d) 4
17. 17.Which of these numbers is divisible by 8? (a) 2248 (b) 2246 (c) 2263 (d) 2251
18. 18.Which of these numbers is divisible by 11? (a) 5282 (b) 5280 (c) 5274 (d) 5293
19. 19.What is the remainder when 188 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
20. 20.What is the remainder when 529 is divided by 7? (a) 3 (b) 5 (c) 4 (d) 6
21. 21.What is the units digit of 7^27? (a) 9 (b) 3 (c) 7 (d) 1
22. 22.Which of these numbers is divisible by 3? (a) 2312 (b) 2309 (c) 2314 (d) 2310
23. 23.What is the remainder when 372 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
24. 24.What is the remainder when 309 is divided by 8? (a) 4 (b) 6 (c) 3 (d) 5
25. 25.Which of these numbers is divisible by 6? (a) 1572 (b) 1573 (c) 1582 (d) 1570
26. 26.What is the units digit of 4^14? (a) 7 (b) 5 (c) 6 (d) 4
27. 27.What is the units digit of 3^31? (a) 7 (b) 3 (c) 9 (d) 1
28. 28.What is the units digit of 2^29? (a) 4 (b) 6 (c) 2 (d) 8
29. 29.What is the remainder when 106 is divided by 3? (a) 1 (b) 2 (c) 0
30. 30.What is the smallest number you must add to 217 to make it divisible by 3? (a) 2 (b) 1
31. 31.What is the units digit of 3^26? (a) 3 (b) 1 (c) 9 (d) 7
32. 32.Which of these numbers is divisible by 9? (a) 1421 (b) 1422 (c) 1423 (d) 1434
33. 33.What is the units digit of 9^37? (a) 8 (b) 1 (c) 9 (d) 7
34. 34.Which of these numbers is divisible by 6? (a) 1004 (b) 1019 (c) 1011 (d) 1008
35. 35.What is the smallest number you must add to 46 to make it divisible by 3? (a) 1 (b) 2
36. 36.What is the smallest number you must add to 799 to make it divisible by 10? (a) 1 (b) 3 (c) 2 (d) 9
37. 37.What is the units digit of 7^36? (a) 9 (b) 3 (c) 1 (d) 7
38. 38.What is the smallest number you must add to 412 to make it divisible by 11? (a) 6 (b) 8 (c) 5 (d) 7
39. 39.What is the remainder when 241 is divided by 3? (a) 2 (b) 0 (c) 1
40. 40.What is the remainder when 508 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 4

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