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

Mellie  Jun 23, 2015

Best Answer 

 #3
avatar+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
avatar+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
avatar+19632 
+15
Best Answer

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
avatar+26750 
+5

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

.

Alan  Jun 23, 2015
 #5
avatar+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

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