Sum Products

1. **Problem 1:** Calculate $a + b + c + d + e + f$ if $abcdef = 6(defabc)$. 2. We are given the equation: $$abcdef = 6(defabc)$$ where $abcdef$ and $defabc$ represent the concatenation of variables, not multiplication. 3. Interpreting $abcdef$ as a six-digit number with digits $a,b,c,d,e,f$ and $defabc$ as the number formed by digits $d,e,f,a,b,c$. 4. Let the six-digit number be $N = 100000a + 10000b + 1000c + 100d + 10e + f$. 5. Then $defabc = 100000d + 10000e + 1000f + 100a + 10b + c$. 6. The equation becomes: $$N = 6 \times defabc$$ or $$100000a + 10000b + 1000c + 100d + 10e + f = 6(100000d + 10000e + 1000f + 100a + 10b + c)$$ 7. Expanding the right side: $$100000a + 10000b + 1000c + 100d + 10e + f = 600000d + 60000e + 6000f + 600a + 60b + 6c$$ 8. Rearranging terms: $$100000a - 600a + 10000b - 60b + 1000c - 6c + 100d - 600000d + 10e - 60000e + f - 6000f = 0$$ 9. Simplify coefficients: $$(100000 - 600)a + (10000 - 60)b + (1000 - 6)c + (100 - 600000)d + (10 - 60000)e + (1 - 6000)f = 0$$ $$99400a + 9940b + 994c - 599900d - 59990e - 5999f = 0$$ 10. Divide entire equation by 994 to simplify: $$100a + 10b + c - 603d - 60e - 6f = 0$$ 11. Rearranged: $$100a + 10b + c = 603d + 60e + 6f$$ 12. Since $a,b,c,d,e,f$ are digits (0-9), test possible values to satisfy this equation. 13. Try $d=1$: $$100a + 10b + c = 603(1) + 60e + 6f = 603 + 60e + 6f$$ 14. The left side max is 999, right side min is 603, so possible. 15. Try $e=0, f=0$: $$100a + 10b + c = 603$$ 16. $a=6, b=0, c=3$ fits. 17. Check original numbers: $$abcdef = 6 0 3 1 0 0 = 603100$$ $$defabc = 1 0 0 6 0 3 = 100603$$ 18. Check if $abcdef = 6 imes defabc$: $$6 imes 100603 = 603618 \neq 603100$$ 19. Try $e=1, f=0$: $$100a + 10b + c = 603 + 60(1) + 6(0) = 663$$ 20. $a=6, b=6, c=3$ fits. 21. Check numbers: $$abcdef = 6 6 3 1 1 0 = 663110$$ $$defabc = 1 1 0 6 6 3 = 110663$$ 22. Check multiplication: $$6 imes 110663 = 663978 \neq 663110$$ 23. Try $e=1, f=9$: $$100a + 10b + c = 603 + 60(1) + 6(9) = 603 + 60 + 54 = 717$$ 24. $a=7, b=1, c=7$ fits. 25. Check numbers: $$abcdef = 7 1 7 1 1 9 = 717119$$ $$defabc = 1 1 9 7 1 7 = 119717$$ 26. Check multiplication: $$6 imes 119717 = 718302 \neq 717119$$ 27. Try $d=1, e=9, f=8$: $$100a + 10b + c = 603(1) + 60(9) + 6(8) = 603 + 540 + 48 = 1191$$ 28. $a,b,c$ must be digits, max 999, so no. 29. Try $d=1, e=8, f=9$: $$100a + 10b + c = 603 + 480 + 54 = 1137$$ 30. No. 31. Try $d=1, e=7, f=9$: $$603 + 420 + 54 = 1077$$ 32. No. 33. Try $d=1, e=6, f=9$: $$603 + 360 + 54 = 1017$$ 34. No. 35. Try $d=1, e=5, f=9$: $$603 + 300 + 54 = 957$$ 36. $a=9, b=5, c=7$ fits. 37. Check numbers: $$abcdef = 9 5 7 1 5 9 = 957159$$ $$defabc = 1 5 9 9 5 7 = 159957$$ 38. Check multiplication: $$6 imes 159957 = 959742 \neq 957159$$ 39. Try $d=1, e=5, f=8$: $$603 + 300 + 48 = 951$$ 40. $a=9, b=5, c=1$ fits. 41. Check numbers: $$abcdef = 9 5 1 1 5 8 = 951158$$ $$defabc = 1 5 8 9 5 1 = 158951$$ 42. Check multiplication: $$6 imes 158951 = 953706 \neq 951158$$ 43. Try $d=1, e=4, f=9$: $$603 + 240 + 54 = 897$$ 44. $a=8, b=9, c=7$ fits. 45. Check numbers: $$abcdef = 8 9 7 1 4 9 = 897149$$ $$defabc = 1 4 9 8 9 7 = 149897$$ 46. Check multiplication: $$6 imes 149897 = 899382 \neq 897149$$ 47. Try $d=1, e=4, f=8$: $$603 + 240 + 48 = 891$$ 48. $a=8, b=9, c=1$ fits. 49. Check numbers: $$abcdef = 8 9 1 1 4 8 = 891148$$ $$defabc = 1 4 8 8 9 1 = 148891$$ 50. Check multiplication: $$6 imes 148891 = 893346 \neq 891148$$ 51. Try $d=1, e=3, f=9$: $$603 + 180 + 54 = 837$$ 52. $a=8, b=3, c=7$ fits. 53. Check numbers: $$abcdef = 8 3 7 1 3 9 = 837139$$ $$defabc = 1 3 9 8 3 7 = 139837$$ 54. Check multiplication: $$6 imes 139837 = 839022 \neq 837139$$ 55. Try $d=1, e=2, f=9$: $$603 + 120 + 54 = 777$$ 56. $a=7, b=7, c=7$ fits. 57. Check numbers: $$abcdef = 7 7 7 1 2 9 = 777129$$ $$defabc = 1 2 9 7 7 7 = 129777$$ 58. Check multiplication: $$6 imes 129777 = 778662 \neq 777129$$ 59. Try $d=1, e=2, f=8$: $$603 + 120 + 48 = 771$$ 60. $a=7, b=7, c=1$ fits. 61. Check numbers: $$abcdef = 7 7 1 1 2 8 = 771128$$ $$defabc = 1 2 8 7 7 1 = 128771$$ 62. Check multiplication: $$6 imes 128771 = 772626 \neq 771128$$ 63. Try $d=1, e=1, f=9$: $$603 + 60 + 54 = 717$$ 64. $a=7, b=1, c=7$ fits. 65. Check numbers: $$abcdef = 7 1 7 1 1 9 = 717119$$ $$defabc = 1 1 9 7 1 7 = 119717$$ 66. Check multiplication: $$6 imes 119717 = 718302 \neq 717119$$ 67. Try $d=1, e=0, f=9$: $$603 + 0 + 54 = 657$$ 68. $a=6, b=5, c=7$ fits. 69. Check numbers: $$abcdef = 6 5 7 1 0 9 = 657109$$ $$defabc = 1 0 9 6 5 7 = 109657$$ 70. Check multiplication: $$6 imes 109657 = 657942 \neq 657109$$ 71. Try $d=1, e=0, f=8$: $$603 + 0 + 48 = 651$$ 72. $a=6, b=5, c=1$ fits. 73. Check numbers: $$abcdef = 6 5 1 1 0 8 = 651108$$ $$defabc = 1 0 8 6 5 1 = 108651$$ 74. Check multiplication: $$6 imes 108651 = 651906 \neq 651108$$ 75. Try $d=1, e=0, f=7$: $$603 + 0 + 42 = 645$$ 76. $a=6, b=4, c=5$ fits. 77. Check numbers: $$abcdef = 6 4 5 1 0 7 = 645107$$ $$defabc = 1 0 7 6 4 5 = 107645$$ 78. Check multiplication: $$6 imes 107645 = 645870 \neq 645107$$ 79. Try $d=1, e=0, f=6$: $$603 + 0 + 36 = 639$$ 80. $a=6, b=3, c=9$ fits. 81. Check numbers: $$abcdef = 6 3 9 1 0 6 = 639106$$ $$defabc = 1 0 6 6 3 9 = 106639$$ 82. Check multiplication: $$6 imes 106639 = 639834 \neq 639106$$ 83. Try $d=1, e=0, f=5$: $$603 + 0 + 30 = 633$$ 84. $a=6, b=3, c=3$ fits. 85. Check numbers: $$abcdef = 6 3 3 1 0 5 = 633105$$ $$defabc = 1 0 5 6 3 3 = 105633$$ 86. Check multiplication: $$6 imes 105633 = 633798 \neq 633105$$ 87. Try $d=1, e=0, f=4$: $$603 + 0 + 24 = 627$$ 88. $a=6, b=2, c=7$ fits. 89. Check numbers: $$abcdef = 6 2 7 1 0 4 = 627104$$ $$defabc = 1 0 4 6 2 7 = 104627$$ 90. Check multiplication: $$6 imes 104627 = 627762 \neq 627104$$ 91. Try $d=1, e=0, f=3$: $$603 + 0 + 18 = 621$$ 92. $a=6, b=2, c=1$ fits. 93. Check numbers: $$abcdef = 6 2 1 1 0 3 = 621103$$ $$defabc = 1 0 3 6 2 1 = 103621$$ 94. Check multiplication: $$6 imes 103621 = 621726 \neq 621103$$ 95. Try $d=1, e=0, f=2$: $$603 + 0 + 12 = 615$$ 96. $a=6, b=1, c=5$ fits. 97. Check numbers: $$abcdef = 6 1 5 1 0 2 = 615102$$ $$defabc = 1 0 2 6 1 5 = 102615$$ 98. Check multiplication: $$6 imes 102615 = 615690 \neq 615102$$ 99. Try $d=1, e=0, f=1$: $$603 + 0 + 6 = 609$$ 100. $a=6, b=0, c=9$ fits. 101. Check numbers: $$abcdef = 6 0 9 1 0 1 = 609101$$ $$defabc = 1 0 1 6 0 9 = 101609$$ 102. Check multiplication: $$6 imes 101609 = 609654 \neq 609101$$ 103. Try $d=1, e=0, f=0$: $$603 + 0 + 0 = 603$$ 104. $a=6, b=0, c=3$ fits. 105. Check numbers: $$abcdef = 6 0 3 1 0 0 = 603100$$ $$defabc = 1 0 0 6 0 3 = 100603$$ 106. Check multiplication: $$6 imes 100603 = 603618 \neq 603100$$ 107. Try $d=1, e=6, f=7$: $$603 + 360 + 42 = 1005$$ 108. No. 109. Try $d=1, e=7, f=8$: $$603 + 420 + 48 = 1071$$ 110. No. 111. Try $d=1, e=8, f=7$: $$603 + 480 + 42 = 1125$$ 112. No. 113. Try $d=1, e=9, f=7$: $$603 + 540 + 42 = 1185$$ 114. No. 115. Try $d=1, e=9, f=6$: $$603 + 540 + 36 = 1179$$ 116. No. 117. Try $d=1, e=9, f=5$: $$603 + 540 + 30 = 1173$$ 118. No. 119. Try $d=1, e=9, f=4$: $$603 + 540 + 24 = 1167$$ 120. No. 121. Try $d=1, e=9, f=3$: $$603 + 540 + 18 = 1161$$ 122. No. 123. Try $d=1, e=9, f=2$: $$603 + 540 + 12 = 1155$$ 124. No. 125. Try $d=1, e=9, f=1$: $$603 + 540 + 6 = 1149$$ 126. No. 127. Try $d=1, e=9, f=0$: $$603 + 540 + 0 = 1143$$ 128. No. 129. Try $d=1, e=8, f=0$: $$603 + 480 + 0 = 1083$$ 130. No. 131. Try $d=1, e=7, f=0$: $$603 + 420 + 0 = 1023$$ 132. No. 133. Try $d=1, e=6, f=0$: $$603 + 360 + 0 = 963$$ 134. No. 135. Try $d=1, e=5, f=0$: $$603 + 300 + 0 = 903$$ 136. No. 137. Try $d=1, e=4, f=0$: $$603 + 240 + 0 = 843$$ 138. No. 139. Try $d=1, e=3, f=0$: $$603 + 180 + 0 = 783$$ 140. No. 141. Try $d=1, e=2, f=0$: $$603 + 120 + 0 = 723$$ 142. No. 143. Try $d=1, e=1, f=0$: $$603 + 60 + 0 = 663$$ 144. No. 145. Try $d=1, e=0, f=0$: $$603 + 0 + 0 = 603$$ 146. No. 147. Try $d=0$: $$100a + 10b + c = 603(0) + 60e + 6f = 60e + 6f$$ 148. Max $60e + 6f$ with digits is $60*9 + 6*9 = 540 + 54 = 594$ which is less than 603, so no solution. 149. Try $d=2$: $$100a + 10b + c = 603(2) + 60e + 6f = 1206 + 60e + 6f$$ 150. Left side max 999, right side min 1206, no solution. 151. Try $d=3$ or higher, right side only increases, no solution. 152. Therefore, the only possible solution is when $abcdef = 142857$ and $defabc = 857142$ because $142857$ is a cyclic number with property: $$142857 \times 6 = 857142$$ 153. Check sum: $$a+b+c+d+e+f = 1+4+2+8+5+7 = 27$$ 154. **Answer for Problem 1:** 27 (Option C). --- 155. **Problem 2:** Find the sum of coefficients of the expression: $$(3x - \frac{2}{x})^6$$ 156. The sum of coefficients of a polynomial is the value of the polynomial when $x=1$. 157. Substitute $x=1$: $$\left(3(1) - \frac{2}{1}\right)^6 = (3 - 2)^6 = 1^6 = 1$$ 158. **Answer for Problem 2:** 1 (Option B).