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# How many numbers between 1000 and 2000 leave a remainder of 3 when divided by 11?

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1136
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+1781

How many numbers between 1000 and 2000 leave a remainder of 3 when divided by 11?

Mellie  Jun 23, 2015

#3
+19632
+15

How many numbers between 1000 and 2000 leave a remainder of 3 when divided by 11 ?

$$\small{ \begin{array}{rcl} n_1 \cdot 11 + 3 &\ge& 1000 \\ n_1 &\ge& \dfrac{1000-3}{11} = 90.6363636364 \\ n_1 &=& 91 \\\\ n_2 \cdot 11 + 3 &\le& 2000 \\ n_2 &\le& \dfrac{2000-3}{11} = 181.545454545 \\ n_2 &=& 181 \\\\ n&=& n_2-n_1+1\\ n&=& 181-91+1\\ \mathbf{n}&\mathbf{=}& \mathbf{91}\\\\ \hline \end{array} }$$

heureka  Jun 23, 2015
#2
+19632
+15

How many numbers between 1000 and 2000 leave a remainder of 3 when divided by 11 ?

$$\small{ \begin{array}{rcl} 1. & 1004 & \text{remainder~} 3 \\ 2. & 1015 & \text{remainder~} 3\\ 3. & 1026 & \text{remainder~} 3\\ 4. & 1037 & \text{remainder~} 3\\ 5. & 1048 & \text{remainder~} 3\\ 6. & 1059 & \text{remainder~} 3\\ 7. & 1070 & \text{remainder~} 3 \\ 8. & 1081 & \text{remainder~} 3\\ 9. & 1092 & \text{remainder~} 3\\ 10. & 1103 & \text{remainder~} 3\\ 11. & 1114 & \text{remainder~} 3\\ 12. & 1125 & \text{remainder~} 3\\ 13. & 1136 & \text{remainder~} 3\\ 14. & 1147 & \text{remainder~} 3\\ 15. & 1158 & \text{remainder~} 3\\ 16. & 1169 & \text{remainder~} 3\\ 17. & 1180 & \text{remainder~} 3\\ 18. & 1191 & \text{remainder~} 3\\ 19. & 1202 & \text{remainder~} 3\\ 20. & 1213 & \text{remainder~} 3\\ 21. & 1224 & \text{remainder~} 3\\ 22. & 1235 & \text{remainder~} 3\\ 23. & 1246 & \text{remainder~} 3\\ 24. & 1257 & \text{remainder~} 3\\ 25. & 1268 & \text{remainder~} 3\\ 26. & 1279 & \text{remainder~} 3\\ 27. & 1290 & \text{remainder~} 3\\ 28. & 1301 & \text{remainder~} 3\\ 29. & 1312 & \text{remainder~} 3\\ 30. & 1323 & \text{remainder~} 3\\ 31. & 1334 & \text{remainder~} 3\\ 32. & 1345 & \text{remainder~} 3\\ 33. & 1356 & \text{remainder~} 3\\ 34. & 1367 & \text{remainder~} 3\\ 35. & 1378 & \text{remainder~} 3\\ 36. & 1389 & \text{remainder~} 3\\ 37. & 1400 & \text{remainder~} 3\\ 38. & 1411 & \text{remainder~} 3\\ 39. & 1422 & \text{remainder~} 3\\ 40. & 1433 & \text{remainder~} 3\\ 41. & 1444 & \text{remainder~} 3\\ 42. & 1455 & \text{remainder~} 3\\ 43. & 1466 & \text{remainder~} 3\\ 44. & 1477 & \text{remainder~} 3\\ 45. & 1488 & \text{remainder~} 3\\ 46. & 1499 & \text{remainder~} 3\\ 47. & 1510 & \text{remainder~} 3\\ 48. & 1521 & \text{remainder~} 3\\ 49. & 1532 & \text{remainder~} 3\\ 50. & 1543 & \text{remainder~} 3\\ 51. & 1554 & \text{remainder~} 3\\ 52. & 1565 & \text{remainder~} 3\\ 53. & 1576 & \text{remainder~} 3\\ 54. & 1587 & \text{remainder~} 3\\ 55. & 1598 & \text{remainder~} 3\\ 56. & 1609 & \text{remainder~} 3\\ 57. & 1620 & \text{remainder~} 3\\ 58. & 1631 & \text{remainder~} 3\\ 59. & 1642 & \text{remainder~} 3\\ 60. & 1653 & \text{remainder~} 3\\ 61. & 1664 & \text{remainder~} 3\\ 62. & 1675 & \text{remainder~} 3\\ 63. & 1686 & \text{remainder~} 3\\ 64. & 1697 & \text{remainder~} 3\\ 65. & 1708 & \text{remainder~} 3\\ 66. & 1719 & \text{remainder~} 3\\ 67. & 1730 & \text{remainder~} 3\\ 68. & 1741 & \text{remainder~} 3\\ 69. & 1752 & \text{remainder~} 3\\ 70. & 1763 & \text{remainder~} 3\\ 71. & 1774 & \text{remainder~} 3\\ 72. & 1785 & \text{remainder~} 3\\ 73. & 1796 & \text{remainder~} 3\\ 74. & 1807 & \text{remainder~} 3\\ 75. & 1818 & \text{remainder~} 3\\ 76. & 1829 & \text{remainder~} 3\\ 77. & 1840 & \text{remainder~} 3\\ 78. & 1851 & \text{remainder~} 3\\ 79. & 1862 & \text{remainder~} 3\\ 80. & 1873 & \text{remainder~} 3\\ 81. & 1884 & \text{remainder~} 3\\ 82. & 1895 & \text{remainder~} 3\\ 83. & 1906 & \text{remainder~} 3\\ 84. & 1917 & \text{remainder~} 3\\ 85. & 1928 & \text{remainder~} 3\\ 86. & 1939 & \text{remainder~} 3\\ 87. & 1950 & \text{remainder~} 3\\ 88. & 1961 & \text{remainder~} 3\\ 89. & 1972 & \text{remainder~} 3\\ 90. & 1983 & \text{remainder~} 3\\ 91. & 1994 & \text{remainder~} 3\\ \end{array} }$$

heureka  Jun 23, 2015
#3
+19632
+15

How many numbers between 1000 and 2000 leave a remainder of 3 when divided by 11 ?

$$\small{ \begin{array}{rcl} n_1 \cdot 11 + 3 &\ge& 1000 \\ n_1 &\ge& \dfrac{1000-3}{11} = 90.6363636364 \\ n_1 &=& 91 \\\\ n_2 \cdot 11 + 3 &\le& 2000 \\ n_2 &\le& \dfrac{2000-3}{11} = 181.545454545 \\ n_2 &=& 181 \\\\ n&=& n_2-n_1+1\\ n&=& 181-91+1\\ \mathbf{n}&\mathbf{=}& \mathbf{91}\\\\ \hline \end{array} }$$

heureka  Jun 23, 2015
#4
+26750
+5

I prefer your second solution heureka; it's much more elegant!

.

Alan  Jun 23, 2015
#5
+92781
+5

Hi Heureka,

Yes Alan Heureka's 2nd answer is more elegant but Mellie may understand the first one much better

I like both!

Melody  Jun 23, 2015