Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 633 · 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 151 to make it divisible by 9? (a) 1 (b) 3 (c) 7 (d) 2
2. 2.Which of these numbers is divisible by 9? (a) 927 (b) 920 (c) 938 (d) 935
3. 3.What is the units digit of 4^57? (a) 6 (b) 3 (c) 4 (d) 5
4. 4.What is the remainder when 302 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 2
5. 5.What is the remainder when 278 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
6. 6.Which of these numbers is divisible by 6? (a) 5396 (b) 5391 (c) 5384 (d) 5388
7. 7.Which of these numbers is divisible by 3? (a) 1583 (b) 1584 (c) 1588 (d) 1585
8. 8.Which of these numbers is divisible by 6? (a) 855 (b) 865 (c) 858 (d) 863
9. 9.What is the remainder when 268 is divided by 10? (a) 2 (b) 8 (c) 9 (d) 7
10. 10.What is the smallest number you must add to 46 to make it divisible by 4? (a) 3 (b) 2 (c) 1
11. 11.Which of these numbers is divisible by 11? (a) 1175 (b) 1178 (c) 1166 (d) 1158
12. 12.What is the remainder when 87 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
13. 13.What is the smallest number you must add to 747 to make it divisible by 11? (a) 10 (b) 3 (c) 2 (d) 1
14. 14.Which of these numbers is divisible by 11? (a) 1361 (b) 1347 (c) 1368 (d) 1353
15. 15.What is the remainder when 287 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
16. 16.Which of these numbers is divisible by 3? (a) 1948 (b) 1947 (c) 1945 (d) 1951
17. 17.Which of these numbers is divisible by 8? (a) 1845 (b) 1852 (c) 1837 (d) 1840
18. 18.What is the remainder when 187 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
19. 19.Which of these numbers is divisible by 3? (a) 1554 (b) 1559 (c) 1556 (d) 1553
20. 20.What is the units digit of 3^25? (a) 3 (b) 1 (c) 9 (d) 7
21. 21.What is the smallest number you must add to 609 to make it divisible by 8? (a) 1 (b) 5 (c) 7 (d) 6
22. 22.What is the remainder when 442 is divided by 7? (a) 2 (b) 6 (c) 1 (d) 0
23. 23.What is the remainder when 482 is divided by 7? (a) 1 (b) 4 (c) 6 (d) 5
24. 24.What is the remainder when 876 is divided by 11? (a) 7 (b) 4 (c) 8 (d) 6
25. 25.What is the smallest number you must add to 73 to make it divisible by 3? (a) 2 (b) 1
26. 26.Which of these numbers is divisible by 8? (a) 6760 (b) 6774 (c) 6756 (d) 6763
27. 27.What is the remainder when 483 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 2
28. 28.What is the smallest number you must add to 546 to make it divisible by 9? (a) 6 (b) 3 (c) 2 (d) 4
29. 29.What is the smallest number you must add to 611 to make it divisible by 10? (a) 7 (b) 8 (c) 9 (d) 1
30. 30.What is the smallest number you must add to 274 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
31. 31.Which of these numbers is divisible by 9? (a) 4265 (b) 4270 (c) 4252 (d) 4257
32. 32.Which of these numbers is divisible by 9? (a) 3876 (b) 3890 (c) 3879 (d) 3884
33. 33.What is the remainder when 335 is divided by 6? (a) 5 (b) 1 (c) 3 (d) 4
34. 34.What is the smallest number you must add to 51 to make it divisible by 4? (a) 3 (b) 2 (c) 1
35. 35.What is the remainder when 129 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
36. 36.What is the units digit of 7^14? (a) 9 (b) 1 (c) 3 (d) 7
37. 37.What is the smallest number you must add to 112 to make it divisible by 9? (a) 4 (b) 7 (c) 6 (d) 5
38. 38.What is the units digit of 7^13? (a) 7 (b) 9 (c) 3 (d) 1
39. 39.Which of these numbers is divisible by 6? (a) 1266 (b) 1276 (c) 1268 (d) 1265
40. 40.What is the remainder when 456 is divided by 7? (a) 6 (b) 0 (c) 1 (d) 2

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