Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 277 · 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.Which of these numbers is divisible by 8? (a) 3394 (b) 3414 (c) 3400 (d) 3407
2. 2.Which of these numbers is divisible by 6? (a) 271 (b) 260 (c) 264 (d) 268
3. 3.What is the remainder when 663 is divided by 10? (a) 7 (b) 2 (c) 4 (d) 3
4. 4.What is the units digit of 8^20? (a) 6 (b) 2 (c) 8 (d) 4
5. 5.What is the remainder when 93 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
6. 6.What is the smallest number you must add to 119 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 2
7. 7.Which of these numbers is divisible by 3? (a) 275 (b) 281 (c) 277 (d) 276
8. 8.What is the remainder when 110 is divided by 9? (a) 7 (b) 2 (c) 1 (d) 3
9. 9.What is the remainder when 70 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
10. 10.What is the smallest number you must add to 520 to make it divisible by 7? (a) 4 (b) 6 (c) 2 (d) 5
11. 11.What is the remainder when 792 is divided by 10? (a) 2 (b) 1 (c) 3 (d) 8
12. 12.What is the remainder when 162 is divided by 7? (a) 1 (b) 2 (c) 6 (d) 0
13. 13.What is the units digit of 7^56? (a) 3 (b) 1 (c) 9 (d) 7
14. 14.What is the units digit of 2^26? (a) 4 (b) 8 (c) 2 (d) 6
15. 15.What is the remainder when 132 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
16. 16.What is the smallest number you must add to 404 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
17. 17.What is the smallest number you must add to 202 to make it divisible by 4? (a) 1 (b) 2 (c) 3
18. 18.What is the remainder when 555 is divided by 10? (a) 6 (b) 7 (c) 5 (d) 4
19. 19.What is the units digit of 4^12? (a) 6 (b) 7 (c) 5 (d) 4
20. 20.What is the smallest number you must add to 555 to make it divisible by 10? (a) 7 (b) 4 (c) 5 (d) 6
21. 21.What is the smallest number you must add to 296 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 10
22. 22.What is the units digit of 2^11? (a) 4 (b) 2 (c) 6 (d) 8
23. 23.What is the smallest number you must add to 230 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 3
24. 24.Which of these numbers is divisible by 6? (a) 3245 (b) 3238 (c) 3247 (d) 3240
25. 25.What is the smallest number you must add to 72 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
26. 26.Which of these numbers is divisible by 6? (a) 522 (b) 526 (c) 517 (d) 529
27. 27.What is the units digit of 3^19? (a) 3 (b) 9 (c) 7 (d) 1
28. 28.What is the remainder when 202 is divided by 6? (a) 3 (b) 5 (c) 2 (d) 4
29. 29.What is the smallest number you must add to 123 to make it divisible by 4? (a) 3 (b) 2 (c) 1
30. 30.Which of these numbers is divisible by 4? (a) 1312 (b) 1313 (c) 1311 (d) 1317
31. 31.What is the remainder when 65 is divided by 3? (a) 0 (b) 1 (c) 2
32. 32.Which of these numbers is divisible by 6? (a) 3955 (b) 3951 (c) 3954 (d) 3962
33. 33.Which of these numbers is divisible by 8? (a) 4298 (b) 4309 (c) 4296 (d) 4290
34. 34.What is the smallest number you must add to 502 to make it divisible by 12? (a) 2 (b) 3 (c) 10 (d) 1
35. 35.What is the smallest number you must add to 153 to make it divisible by 11? (a) 10 (b) 2 (c) 1 (d) 3
36. 36.Which of these numbers is divisible by 11? (a) 3104 (b) 3088 (c) 3091 (d) 3100
37. 37.What is the units digit of 4^42? (a) 4 (b) 5 (c) 7 (d) 6
38. 38.What is the remainder when 76 is divided by 5? (a) 1 (b) 0 (c) 2 (d) 4
39. 39.What is the units digit of 8^45? (a) 8 (b) 6 (c) 4 (d) 2
40. 40.What is the units digit of 7^22? (a) 9 (b) 7 (c) 3 (d) 1

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