Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 997 · 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 7^40? (a) 9 (b) 3 (c) 1 (d) 7
2. 2.What is the smallest number you must add to 234 to make it divisible by 12? (a) 8 (b) 5 (c) 6 (d) 7
3. 3.What is the remainder when 97 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
4. 4.What is the smallest number you must add to 221 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
5. 5.What is the units digit of 4^49? (a) 5 (b) 4 (c) 3 (d) 6
6. 6.What is the smallest number you must add to 818 to make it divisible by 11? (a) 6 (b) 7 (c) 4 (d) 8
7. 7.What is the units digit of 2^41? (a) 4 (b) 8 (c) 6 (d) 2
8. 8.What is the units digit of 3^14? (a) 9 (b) 3 (c) 1 (d) 7
9. 9.What is the remainder when 244 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 6
10. 10.What is the units digit of 2^46? (a) 8 (b) 2 (c) 6 (d) 4
11. 11.What is the units digit of 2^49? (a) 8 (b) 6 (c) 2 (d) 4
12. 12.What is the units digit of 9^11? (a) 1 (b) 9 (c) 8 (d) 7
13. 13.What is the units digit of 8^23? (a) 6 (b) 4 (c) 2 (d) 8
14. 14.What is the units digit of 7^42? (a) 7 (b) 1 (c) 3 (d) 9
15. 15.What is the units digit of 4^15? (a) 5 (b) 4 (c) 6 (d) 3
16. 16.What is the smallest number you must add to 129 to make it divisible by 4? (a) 2 (b) 1 (c) 3
17. 17.What is the remainder when 285 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 7
18. 18.What is the smallest number you must add to 303 to make it divisible by 4? (a) 2 (b) 3 (c) 1
19. 19.What is the smallest number you must add to 197 to make it divisible by 4? (a) 1 (b) 2 (c) 3
20. 20.What is the units digit of 7^41? (a) 9 (b) 7 (c) 1 (d) 3
21. 21.Which of these numbers is divisible by 9? (a) 5301 (b) 5293 (c) 5316 (d) 5308
22. 22.What is the remainder when 173 is divided by 9? (a) 3 (b) 2 (c) 7 (d) 1
23. 23.What is the remainder when 230 is divided by 3? (a) 2 (b) 1 (c) 0
24. 24.What is the units digit of 4^26? (a) 4 (b) 7 (c) 5 (d) 6
25. 25.Which of these numbers is divisible by 6? (a) 5273 (b) 5281 (c) 5274 (d) 5276
26. 26.What is the remainder when 206 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
27. 27.Which of these numbers is divisible by 3? (a) 2555 (b) 2557 (c) 2552 (d) 2553
28. 28.What is the units digit of 9^45? (a) 7 (b) 8 (c) 1 (d) 9
29. 29.What is the remainder when 472 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 3
30. 30.What is the units digit of 9^52? (a) 9 (b) 2 (c) 0 (d) 1
31. 31.What is the smallest number you must add to 272 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
32. 32.What is the units digit of 7^58? (a) 3 (b) 9 (c) 1 (d) 7
33. 33.What is the units digit of 9^48? (a) 0 (b) 9 (c) 2 (d) 1
34. 34.What is the units digit of 4^37? (a) 6 (b) 5 (c) 4 (d) 3
35. 35.Which of these numbers is divisible by 4? (a) 1125 (b) 1128 (c) 1134 (d) 1130
36. 36.What is the remainder when 67 is divided by 6? (a) 2 (b) 0 (c) 1 (d) 5
37. 37.What is the remainder when 95 is divided by 9? (a) 6 (b) 5 (c) 4 (d) 7
38. 38.Which of these numbers is divisible by 9? (a) 7275 (b) 7284 (c) 7292 (d) 7281
39. 39.What is the units digit of 2^31? (a) 8 (b) 4 (c) 6 (d) 2
40. 40.What is the units digit of 8^14? (a) 2 (b) 6 (c) 4 (d) 8

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