Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 627 · 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 4^39? (a) 3 (b) 6 (c) 4 (d) 5
2. 2.What is the units digit of 8^47? (a) 2 (b) 4 (c) 6 (d) 8
3. 3.Which of these numbers is divisible by 4? (a) 2431 (b) 2426 (c) 2424 (d) 2421
4. 4.Which of these numbers is divisible by 11? (a) 7626 (b) 7634 (c) 7651 (d) 7638
5. 5.What is the units digit of 4^49? (a) 4 (b) 6 (c) 3 (d) 5
6. 6.Which of these numbers is divisible by 3? (a) 1219 (b) 1221 (c) 1225 (d) 1223
7. 7.What is the smallest number you must add to 107 to make it divisible by 3? (a) 2 (b) 1
8. 8.Which of these numbers is divisible by 4? (a) 2477 (b) 2480 (c) 2487 (d) 2483
9. 9.What is the smallest number you must add to 372 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
10. 10.What is the smallest number you must add to 623 to make it divisible by 8? (a) 1 (b) 2 (c) 7 (d) 3
11. 11.Which of these numbers is divisible by 6? (a) 4359 (b) 4349 (c) 4350 (d) 4354
12. 12.What is the smallest number you must add to 669 to make it divisible by 11? (a) 9 (b) 1 (c) 3 (d) 2
13. 13.What is the smallest number you must add to 378 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
14. 14.What is the remainder when 206 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
15. 15.What is the smallest number you must add to 133 to make it divisible by 3? (a) 2 (b) 1
16. 16.Which of these numbers is divisible by 3? (a) 2206 (b) 2202 (c) 2200 (d) 2203
17. 17.What is the remainder when 802 is divided by 10? (a) 3 (b) 8 (c) 2 (d) 1
18. 18.Which of these numbers is divisible by 9? (a) 3396 (b) 3384 (c) 3392 (d) 3381
19. 19.Which of these numbers is divisible by 4? (a) 3212 (b) 3217 (c) 3214 (d) 3209
20. 20.What is the units digit of 4^15? (a) 3 (b) 5 (c) 6 (d) 4
21. 21.What is the units digit of 2^15? (a) 6 (b) 8 (c) 2 (d) 4
22. 22.Which of these numbers is divisible by 3? (a) 372 (b) 376 (c) 370 (d) 373
23. 23.What is the smallest number you must add to 56 to make it divisible by 3? (a) 2 (b) 1
24. 24.What is the remainder when 293 is divided by 8? (a) 6 (b) 5 (c) 3 (d) 4
25. 25.What is the units digit of 8^41? (a) 2 (b) 8 (c) 6 (d) 4
26. 26.What is the remainder when 412 is divided by 7? (a) 1 (b) 5 (c) 4 (d) 6
27. 27.What is the units digit of 9^45? (a) 8 (b) 7 (c) 1 (d) 9
28. 28.What is the remainder when 302 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
29. 29.What is the smallest number you must add to 429 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 6
30. 30.What is the units digit of 9^28? (a) 2 (b) 1 (c) 0 (d) 9
31. 31.What is the remainder when 510 is divided by 11? (a) 5 (b) 7 (c) 4 (d) 3
32. 32.What is the units digit of 9^16? (a) 9 (b) 0 (c) 2 (d) 1
33. 33.What is the units digit of 4^22? (a) 7 (b) 4 (c) 6 (d) 5
34. 34.Which of these numbers is divisible by 8? (a) 4128 (b) 4130 (c) 4138 (d) 4123
35. 35.What is the remainder when 145 is divided by 6? (a) 2 (b) 1 (c) 5 (d) 0
36. 36.What is the smallest number you must add to 278 to make it divisible by 4? (a) 3 (b) 1 (c) 2
37. 37.What is the units digit of 2^58? (a) 2 (b) 6 (c) 4 (d) 8
38. 38.What is the remainder when 64 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
39. 39.What is the units digit of 8^44? (a) 4 (b) 8 (c) 2 (d) 6
40. 40.Which of these numbers is divisible by 6? (a) 4660 (b) 4646 (c) 4653 (d) 4650

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad