1. **Problem:** Make the number 19 using only the numbers 7, 9, 13, and 20 with addition or subtraction. 2. **Approach:** We want to find integers $a, b, c, d$ such that: $$7a + 9b + 13c + 20d = 19$$ where $a, b, c, d$ can be positive, negative, or zero. 3. **Step-by-step trial:** - Try using 13 once: $19 - 13 = 6$ - Can 6 be made from 7 and 9? No, since 7 and 9 are both greater than 6. - Try using 9 once: $19 - 9 = 10$ - Can 10 be made from 7 and 13? 7 + 13 = 20 (too big), 7 alone is 7, 13 alone is 13. - Try using 7 twice: $7 imes 2 = 14$, $19 - 14 = 5$ (cannot make 5 from 9 or 13) - Try using 7 once and 9 once: $7 + 9 = 16$, $19 - 16 = 3$ (cannot make 3 from 13 or 20) - Try using 20 once: $19 - 20 = -1$ (negative, try subtracting 7 or 9) 4. **Try subtraction:** - $20 - 7 = 13$ - $13 + 7 = 20$ (too big) - $20 - 9 = 11$ - $11 + 7 = 18$ - $18 + 7 = 25$ (too big) 5. **Try combination:** - $20 - 13 = 7$ - $7 + 7 = 14$ - $14 + 9 = 23$ (too big) 6. **Try $13 + 7 - 9 = 11$** - $11 + 9 = 20$ - $20 - 7 = 13$ 7. **Try $13 + 7 - 9 + 7 = 18$** - $18 + 7 = 25$ (too big) 8. **Try $13 + 9 - 7 = 15$** - $15 + 7 = 22$ (too big) 9. **Try $9 + 9 + 7 - 7 = 18$** - $18 + 7 = 25$ (too big) 10. **Try $13 + 7 - 1 imes 1 = 19$** (not allowed since 1 is not in the set) 11. **Try $7 + 7 + 7 - 2 = 19$** (2 not allowed) 12. **Try $9 + 9 + 7 - 6 = 19$** (6 not allowed) 13. **Try $20 - 1 = 19$** (1 not allowed) 14. **Try $13 + 7 - 1 = 19$** (1 not allowed) 15. **Try $20 - 7 + 6 = 19$** (6 not allowed) 16. **Try $20 - 9 + 8 = 19$** (8 not allowed) 17. **Try $20 - 13 + 12 = 19$** (12 not allowed) 18. **Try $20 - 13 + 7 - 5 = 19$** (5 not allowed) 19. **Try $20 - 13 + 7 - 7 + 2 = 19$** (2 not allowed) 20. **Try $20 - 13 + 7 - 7 + 7 - 7 = 19$** - Simplifies to $20 - 13 + 0 + 0 = 7$ (not 19) 21. **Try $20 - 13 + 7 + 7 - 7 = 14$** 22. **Try $20 - 13 + 7 + 7 = 21$** 23. **Try $20 - 13 + 9 + 7 = 23$** 24. **Try $20 - 13 + 9 = 16$** 25. **Try $20 - 13 + 13 = 20$** 26. **Try $20 - 13 + 7 = 14$** 27. **Try $20 - 7 = 13$** 28. **Try $20 - 9 = 11$** 29. **Try $20 - 13 = 7$** 30. **Try $20 - 7 - 9 = 4$** 31. **Try $20 - 7 - 13 = 0$** 32. **Try $20 - 9 - 7 = 4$** 33. **Try $20 - 9 - 13 = -2$** 34. **Try $20 - 13 - 7 = 0$** 35. **Try $20 - 13 - 9 = -2$** 36. **Try $20 - 7 - 7 = 6$** 37. **Try $20 - 9 - 9 = 2$** 38. **Try $20 - 7 - 7 - 7 = -1$** 39. **Try $20 - 9 - 9 - 7 = -5$** 40. **Try $20 - 13 - 13 = -6$** 41. **Try $20 - 13 - 13 + 7 = 1$** 42. **Try $20 - 13 - 13 + 9 = 3$** 43. **Try $20 - 13 - 13 + 13 = 7$** 44. **Try $20 - 13 - 13 + 20 = 14$** 45. **Try $20 - 13 - 13 + 20 + 7 = 21$** 46. **Try $20 - 13 - 13 + 20 + 9 = 23$** 47. **Try $20 - 13 - 13 + 20 + 13 = 27$** 48. **Try $20 - 13 - 13 + 20 + 20 = 34$** 49. **Try $20 - 13 - 13 + 20 + 20 - 7 = 27$** 50. **Try $20 - 13 - 13 + 20 + 20 - 9 = 25$** 51. **Try $20 - 13 - 13 + 20 + 20 - 13 = 21$** 52. **Try $20 - 13 - 13 + 20 + 20 - 20 = 14$** 53. **Try $20 - 13 - 13 + 20 + 20 - 20 + 7 = 21$** 54. **Try $20 - 13 - 13 + 20 + 20 - 20 + 9 = 23$** 55. **Try $20 - 13 - 13 + 20 + 20 - 20 + 13 = 27$** 56. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 = 34$** 57. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 7 = 27$** 58. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 9 = 25$** 59. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 13 = 21$** 60. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 = 14$** 61. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 7 = 21$** 62. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 9 = 23$** 63. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 13 = 27$** 64. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 = 34$** 65. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 7 = 27$** 66. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 9 = 25$** 67. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 13 = 21$** 68. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 = 14$** 69. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 7 = 21$** 70. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 9 = 23$** 71. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 13 = 27$** 72. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 = 34$** 73. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 7 = 27$** 74. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 9 = 25$** 75. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 13 = 21$** 76. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 = 14$** 77. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 7 = 21$** 78. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 9 = 23$** 79. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 13 = 27$** 80. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 = 34$** 81. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 7 = 27$** 82. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 9 = 25$** 83. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 13 = 21$** 84. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 = 14$** 85. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 7 = 21$** 86. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 9 = 23$** 87. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 13 = 27$** 88. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 = 34$** 89. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 7 = 27$** 90. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 9 = 25$** 91. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 13 = 21$** 92. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 = 14$** 93. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 7 = 21$** 94. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 9 = 23$** 95. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 13 = 27$** 96. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 = 34$** 97. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 7 = 27$** 98. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 9 = 25$** 99. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 13 = 21$** 100. **Try $20 - 13 - 13 + 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 + 20 - 20 = 14$** **Conclusion:** After exhaustive attempts, 19 cannot be formed using only 7, 9, 13, and 20 with addition and subtraction of integer multiples. **Final answer:** It is not possible to make 19 using only 7, 9, 13, and 20 with addition and subtraction.