Letter N Digit 63B914

1. **State the problem:** We have the addition: $$AMC + AMC + AMC + AMC + AMC + AMC + AMC + AMC = FUN$$ where each letter represents a different non-zero digit. 2. **Rewrite the sum:** Since $AMC$ is added 8 times, this is: $$8 \times AMC = FUN$$ 3. **Express $AMC$ and $FUN$ as numbers:** Let $A$, $M$, $C$, $F$, $U$, and $N$ be digits. Then: $$AMC = 100A + 10M + C$$ $$FUN = 100F + 10U + N$$ 4. **Set up the equation:** $$8 \times (100A + 10M + C) = 100F + 10U + N$$ 5. **Analyze constraints:** - $A$, $M$, $C$, $F$, $U$, $N$ are distinct non-zero digits. - $8 \times AMC$ is a three-digit number $FUN$. 6. **Try to find $A$:** Since $8 \times AMC$ is three digits, $AMC$ must be between 100 and 124 (because $8 \times 125 = 1000$ which is four digits). 7. **Try $A=1$:** Then $AMC = 100 + 10M + C$. Multiply by 8: $$8 \times (100 + 10M + C) = 800 + 80M + 8C = 100F + 10U + N$$ 8. **Check possible values:** Since $F$ is the hundreds digit of the product, $F=8$. 9. **Rewrite:** $$800 + 80M + 8C = 800 + 10U + N$$ So: $$80M + 8C = 10U + N$$ 10. **Simplify by dividing by 2:** $$40M + 4C = 5U + \frac{N}{2}$$ Since $N$ must be an integer digit, $N$ must be even. 11. **Try values for $M$ and $C$ to satisfy:** We want $10U + N = 80M + 8C$. 12. **Try $M=3$, $C=4$:** $$80 \times 3 + 8 \times 4 = 240 + 32 = 272$$ So $10U + N = 272$ which is impossible since $10U + N$ is two digits. 13. **Try $M=2$, $C=3$:** $$80 \times 2 + 8 \times 3 = 160 + 24 = 184$$ So $10U + N = 184$ impossible. 14. **Try $M=1$, $C=6$:** $$80 \times 1 + 8 \times 6 = 80 + 48 = 128$$ $10U + N = 128$ impossible. 15. **Try $M=0$, $C=5$:** $M$ cannot be zero (non-zero digits). 16. **Try $M=4$, $C=6$:** $$80 \times 4 + 8 \times 6 = 320 + 48 = 368$$ $10U + N = 368$ impossible. 17. **Try $M=1$, $C=7$:** $$80 + 56 = 136$$ $10U + N = 136$ impossible. 18. **Try $M=2$, $C=1$:** $$160 + 8 = 168$$ $10U + N = 168$ impossible. 19. **Try $M=3$, $C=2$:** $$240 + 16 = 256$$ $10U + N = 256$ impossible. 20. **Try $M=4$, $C=1$:** $$320 + 8 = 328$$ $10U + N = 328$ impossible. 21. **Try $M=5$, $C=2$:** $$400 + 16 = 416$$ $10U + N = 416$ impossible. 22. **Try $M=6$, $C=3$:** $$480 + 24 = 504$$ $10U + N = 504$ impossible. 23. **Try $M=7$, $C=4$:** $$560 + 32 = 592$$ $10U + N = 592$ impossible. 24. **Try $M=8$, $C=5$:** $$640 + 40 = 680$$ $10U + N = 680$ impossible. 25. **Try $M=9$, $C=6$:** $$720 + 48 = 768$$ $10U + N = 768$ impossible. 26. **Try $A=1$, $M=2$, $C=4$:** $$AMC = 124$$ $$8 \times 124 = 992$$ So $FUN = 992$ 27. **Check digits:** $F=9$, $U=9$, $N=2$ Digits are $A=1$, $M=2$, $C=4$, $F=9$, $U=9$, $N=2$ $U$ and $F$ are both 9, which violates distinct digits. 28. **Try $A=1$, $M=3$, $C=5$:** $$AMC = 135$$ $$8 \times 135 = 1080$$ four digits, invalid. 29. **Try $A=1$, $M=1$, $C=8$:** $$AMC = 118$$ $$8 \times 118 = 944$$ $F=9$, $U=4$, $N=4$ repeated digits, invalid. 30. **Try $A=1$, $M=4$, $C=7$:** $$AMC = 147$$ $$8 \times 147 = 1176$$ four digits, invalid. 31. **Try $A=1$, $M=5$, $C=3$:** $$AMC = 153$$ $$8 \times 153 = 1224$$ four digits, invalid. 32. **Try $A=1$, $M=6$, $C=2$:** $$AMC = 162$$ $$8 \times 162 = 1296$$ four digits, invalid. 33. **Try $A=1$, $M=7$, $C=3$:** $$AMC = 173$$ $$8 \times 173 = 1384$$ four digits, invalid. 34. **Try $A=1$, $M=8$, $C=4$:** $$AMC = 184$$ $$8 \times 184 = 1472$$ four digits, invalid. 35. **Try $A=1$, $M=9$, $C=5$:** $$AMC = 195$$ $$8 \times 195 = 1560$$ four digits, invalid. 36. **Try $A=2$:** $AMC$ between 200 and 249 (since $8 \times 250 = 2000$ four digits). 37. **Try $A=2$, $M=1$, $C=7$:** $$AMC = 217$$ $$8 \times 217 = 1736$$ four digits, invalid. 38. **Try $A=2$, $M=3$, $C=4$:** $$AMC = 234$$ $$8 \times 234 = 1872$$ four digits, invalid. 39. **Try $A=2$, $M=4$, $C=5$:** $$AMC = 245$$ $$8 \times 245 = 1960$$ four digits, invalid. 40. **Try $A=3$:** $AMC$ between 300 and 312 (since $8 \times 313 = 2504$ four digits). 41. **Try $A=3$, $M=1$, $C=2$:** $$AMC = 312$$ $$8 \times 312 = 2496$$ four digits, invalid. 42. **Try $A=3$, $M=0$, $C=1$:** $M=0$ invalid (non-zero digits). 43. **Try $A=4$ or higher:** $8 \times 400 = 3200$ four digits, invalid. 44. **Conclusion:** The only valid $AMC$ is $124$ with $FUN=992$ but digits $F$ and $U$ are the same. 45. **Reconsider the problem:** Since $F$ and $U$ must be distinct, try to find another approach. 46. **Try to find $N$ from the units digit:** Units digit of $8 \times C = N$ (mod 10). 47. **Try possible $C$ and $N$ pairs:** For $C=1$, $8 \times 1=8$, so $N=8$. For $C=2$, $8 \times 2=16$, $N=6$. For $C=3$, $8 \times 3=24$, $N=4$. For $C=4$, $8 \times 4=32$, $N=2$. For $C=5$, $8 \times 5=40$, $N=0$ invalid (zero digit). For $C=6$, $8 \times 6=48$, $N=8$. For $C=7$, $8 \times 7=56$, $N=6$. For $C=8$, $8 \times 8=64$, $N=4$. For $C=9$, $8 \times 9=72$, $N=2$. 48. **Try $N=2$ with $C=4$ or $9$:** Try $C=4$, $N=2$. 49. **Try $A=1$, $M=3$, $C=4$:** $$AMC = 134$$ $$8 \times 134 = 1072$$ four digits, invalid. 50. **Try $A=1$, $M=2$, $C=4$:** $$AMC = 124$$ $$8 \times 124 = 992$$ three digits, $N=2$ matches. 51. **Check digits:** $A=1$, $M=2$, $C=4$, $F=9$, $U=9$, $N=2$ $F$ and $U$ are the same digit, invalid. 52. **Try $N=4$ with $C=3$ or $8$:** Try $C=3$, $N=4$. 53. **Try $A=1$, $M=5$, $C=3$:** $$AMC = 153$$ $$8 \times 153 = 1224$$ four digits, invalid. 54. **Try $A=1$, $M=6$, $C=3$:** $$AMC = 163$$ $$8 \times 163 = 1304$$ four digits, invalid. 55. **Try $N=6$ with $C=2$ or $7$:** Try $C=2$, $N=6$. 56. **Try $A=1$, $M=1$, $C=2$:** $$AMC = 112$$ $$8 \times 112 = 896$$ $F=8$, $U=9$, $N=6$ 57. **Check digits distinctness:** $A=1$, $M=1$ repeated digits invalid. 58. **Try $A=1$, $M=3$, $C=2$:** $$AMC = 132$$ $$8 \times 132 = 1056$$ four digits, invalid. 59. **Try $A=1$, $M=4$, $C=2$:** $$AMC = 142$$ $$8 \times 142 = 1136$$ four digits, invalid. 60. **Try $A=1$, $M=5$, $C=2$:** $$AMC = 152$$ $$8 \times 152 = 1216$$ four digits, invalid. 61. **Try $N=8$ with $C=1$ or $6$:** Try $C=1$, $N=8$. 62. **Try $A=1$, $M=2$, $C=1$:** $$AMC = 121$$ $$8 \times 121 = 968$$ $F=9$, $U=6$, $N=8$ 63. **Check digits distinctness:** $A=1$, $M=2$, $C=1$ repeated digits invalid. 64. **Try $A=1$, $M=3$, $C=1$:** $$AMC = 131$$ $$8 \times 131 = 1048$$ four digits, invalid. 65. **Try $A=1$, $M=4$, $C=1$:** $$AMC = 141$$ $$8 \times 141 = 1128$$ four digits, invalid. 66. **Try $A=1$, $M=5$, $C=1$:** $$AMC = 151$$ $$8 \times 151 = 1208$$ four digits, invalid. 67. **Try $A=1$, $M=6$, $C=1$:** $$AMC = 161$$ $$8 \times 161 = 1288$$ four digits, invalid. 68. **Try $A=1$, $M=7$, $C=1$:** $$AMC = 171$$ $$8 \times 171 = 1368$$ four digits, invalid. 69. **Try $A=1$, $M=8$, $C=1$:** $$AMC = 181$$ $$8 \times 181 = 1448$$ four digits, invalid. 70. **Try $A=1$, $M=9$, $C=1$:** $$AMC = 191$$ $$8 \times 191 = 1528$$ four digits, invalid. 71. **Try $N=4$ with $C=8$:** $$8 \times 8 = 64$$ units digit $4$ matches $N=4$. 72. **Try $A=1$, $M=2$, $C=8$:** $$AMC = 128$$ $$8 \times 128 = 1024$$ four digits, invalid. 73. **Try $A=1$, $M=3$, $C=8$:** $$AMC = 138$$ $$8 \times 138 = 1104$$ four digits, invalid. 74. **Try $A=1$, $M=4$, $C=8$:** $$AMC = 148$$ $$8 \times 148 = 1184$$ four digits, invalid. 75. **Try $A=1$, $M=5$, $C=8$:** $$AMC = 158$$ $$8 \times 158 = 1264$$ four digits, invalid. 76. **Try $A=1$, $M=6$, $C=8$:** $$AMC = 168$$ $$8 \times 168 = 1344$$ four digits, invalid. 77. **Try $A=1$, $M=7$, $C=8$:** $$AMC = 178$$ $$8 \times 178 = 1424$$ four digits, invalid. 78. **Try $A=1$, $M=8$, $C=8$:** $M$ and $C$ same digit invalid. 79. **Try $A=1$, $M=9$, $C=8$:** $$AMC = 198$$ $$8 \times 198 = 1584$$ four digits, invalid. 80. **Try $N=2$ with $C=4$ or $9$ again:** $C=4$, $N=2$ tried before. 81. **Try $A=1$, $M=2$, $C=4$:** $$AMC = 124$$ $$8 \times 124 = 992$$ $F=9$, $U=9$, $N=2$ repeated digits invalid. 82. **Try $C=9$, $N=2$:** $$8 \times 9 = 72$$ units digit 2 matches $N=2$. 83. **Try $A=1$, $M=2$, $C=9$:** $$AMC = 129$$ $$8 \times 129 = 1032$$ four digits, invalid. 84. **Try $A=1$, $M=3$, $C=9$:** $$AMC = 139$$ $$8 \times 139 = 1112$$ four digits, invalid. 85. **Try $A=1$, $M=4$, $C=9$:** $$AMC = 149$$ $$8 \times 149 = 1192$$ four digits, invalid. 86. **Try $A=1$, $M=5$, $C=9$:** $$AMC = 159$$ $$8 \times 159 = 1272$$ four digits, invalid. 87. **Try $A=1$, $M=6$, $C=9$:** $$AMC = 169$$ $$8 \times 169 = 1352$$ four digits, invalid. 88. **Try $A=1$, $M=7$, $C=9$:** $$AMC = 179$$ $$8 \times 179 = 1432$$ four digits, invalid. 89. **Try $A=1$, $M=8$, $C=9$:** $$AMC = 189$$ $$8 \times 189 = 1512$$ four digits, invalid. 90. **Try $A=1$, $M=9$, $C=9$:** $M$ and $C$ same digit invalid. 91. **Try $N=8$ with $C=1$ or $6$ again:** $C=6$, $N=8$. 92. **Try $A=1$, $M=2$, $C=6$:** $$AMC = 126$$ $$8 \times 126 = 1008$$ four digits, invalid. 93. **Try $A=1$, $M=3$, $C=6$:** $$AMC = 136$$ $$8 \times 136 = 1088$$ four digits, invalid. 94. **Try $A=1$, $M=4$, $C=6$:** $$AMC = 146$$ $$8 \times 146 = 1168$$ four digits, invalid. 95. **Try $A=1$, $M=5$, $C=6$:** $$AMC = 156$$ $$8 \times 156 = 1248$$ four digits, invalid. 96. **Try $A=1$, $M=6$, $C=6$:** $M$ and $C$ same digit invalid. 97. **Try $A=1$, $M=7$, $C=6$:** $$AMC = 176$$ $$8 \times 176 = 1408$$ four digits, invalid. 98. **Try $A=1$, $M=8$, $C=6$:** $$AMC = 186$$ $$8 \times 186 = 1488$$ four digits, invalid. 99. **Try $A=1$, $M=9$, $C=6$:** $$AMC = 196$$ $$8 \times 196 = 1568$$ four digits, invalid. 100. **Try $N=4$ with $C=3$ or $8$ again:** $C=3$, $N=4$ tried before. 101. **Try $A=1$, $M=2$, $C=3$:** $$AMC = 123$$ $$8 \times 123 = 984$$ $F=9$, $U=8$, $N=4$ 102. **Check digits distinctness:** $A=1$, $M=2$, $C=3$, $F=9$, $U=8$, $N=4$ all distinct and non-zero. 103. **Answer:** The digit $N$ represents is $4$. **Final answer:** $\boxed{4}$