Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 931 · 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 remainder when 304 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
2. 2.What is the smallest number you must add to 284 to make it divisible by 10? (a) 7 (b) 6 (c) 4 (d) 5
3. 3.What is the smallest number you must add to 537 to make it divisible by 8? (a) 5 (b) 7 (c) 6 (d) 1
4. 4.What is the smallest number you must add to 59 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
5. 5.What is the units digit of 3^27? (a) 1 (b) 9 (c) 7 (d) 3
6. 6.Which of these numbers is divisible by 9? (a) 680 (b) 660 (c) 666 (d) 670
7. 7.What is the units digit of 7^46? (a) 3 (b) 7 (c) 1 (d) 9
8. 8.What is the units digit of 3^36? (a) 9 (b) 7 (c) 1 (d) 3
9. 9.What is the units digit of 4^14? (a) 6 (b) 4 (c) 5 (d) 7
10. 10.Which of these numbers is divisible by 8? (a) 3945 (b) 3936 (c) 3938 (d) 3932
11. 11.What is the units digit of 7^57? (a) 9 (b) 1 (c) 7 (d) 3
12. 12.What is the remainder when 406 is divided by 8? (a) 5 (b) 2 (c) 6 (d) 7
13. 13.What is the units digit of 7^26? (a) 7 (b) 3 (c) 1 (d) 9
14. 14.Which of these numbers is divisible by 8? (a) 2144 (b) 2143 (c) 2157 (d) 2150
15. 15.Which of these numbers is divisible by 3? (a) 2089 (b) 2086 (c) 2084 (d) 2085
16. 16.What is the units digit of 2^30? (a) 8 (b) 4 (c) 6 (d) 2
17. 17.Which of these numbers is divisible by 4? (a) 627 (b) 619 (c) 623 (d) 620
18. 18.What is the units digit of 8^11? (a) 6 (b) 8 (c) 2 (d) 4
19. 19.What is the smallest number you must add to 191 to make it divisible by 9? (a) 8 (b) 6 (c) 7 (d) 2
20. 20.What is the units digit of 9^44? (a) 2 (b) 1 (c) 9 (d) 0
21. 21.What is the units digit of 9^35? (a) 8 (b) 7 (c) 9 (d) 1
22. 22.What is the remainder when 163 is divided by 11? (a) 2 (b) 9 (c) 10 (d) 8
23. 23.What is the smallest number you must add to 467 to make it divisible by 9? (a) 3 (b) 8 (c) 2 (d) 1
24. 24.What is the units digit of 2^46? (a) 6 (b) 4 (c) 2 (d) 8
25. 25.Which of these numbers is divisible by 6? (a) 1931 (b) 1923 (c) 1934 (d) 1926
26. 26.What is the units digit of 4^36? (a) 7 (b) 4 (c) 5 (d) 6
27. 27.What is the smallest number you must add to 146 to make it divisible by 3? (a) 1 (b) 2
28. 28.What is the remainder when 351 is divided by 7? (a) 0 (b) 1 (c) 2 (d) 6
29. 29.What is the units digit of 2^47? (a) 6 (b) 2 (c) 4 (d) 8
30. 30.What is the remainder when 443 is divided by 11? (a) 3 (b) 8 (c) 2 (d) 4
31. 31.Which of these numbers is divisible by 6? (a) 3448 (b) 3450 (c) 3457 (d) 3453
32. 32.What is the units digit of 8^48? (a) 8 (b) 2 (c) 6 (d) 4
33. 33.What is the smallest number you must add to 104 to make it divisible by 10? (a) 7 (b) 6 (c) 4 (d) 5
34. 34.What is the units digit of 3^58? (a) 7 (b) 1 (c) 9 (d) 3
35. 35.What is the remainder when 761 is divided by 11? (a) 3 (b) 2 (c) 1 (d) 9
36. 36.What is the remainder when 483 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
37. 37.Which of these numbers is divisible by 9? (a) 3123 (b) 3128 (c) 3120 (d) 3139
38. 38.What is the remainder when 587 is divided by 9? (a) 1 (b) 7 (c) 3 (d) 2
39. 39.What is the remainder when 110 is divided by 3? (a) 0 (b) 1 (c) 2
40. 40.Which of these numbers is divisible by 6? (a) 1001 (b) 996 (c) 1005 (d) 993

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