For example, if you have two objects, one with weight 5 and one with weight 7,...

  

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For example, if you have two objects, one with weight 5 and one with weight 7, you can measure items of weight 5, 7, or 12. You can also measure items of weight 2, by placing 7 on one side and 5 on the side which contains the object to be weighed. If the scale balances, you know that the item has weight 2 since 5+2=7. Your goal is to measure items of weight N, where N is any whole number between 1 and 40. (a) Suppose you have six objects, with weights 1, 2, 4, 8, 16, 32. Clearly explain how you can use the balance scale to measure an item of weight N, where N is any whole number between 1 and 40. (b) Suppose you have four objects, with weights 1, 3, 9, 27. Clearly explain how you can use the balance scale to measure an item of weight N, where N is any whole number between 1 and 40. (c) Suppose you have three objects, with weights A, B, C. Determine whether there exists a set of three weights A, B, C for which you can use the balance scale to measure an item of weight N, where N is any whole number between 1 and 40. If these three weights exist, determine the values of A, B, C. If these three weights do not ex- ist, clearly explain why this problem is impossible to solve. For example, if you have two objects, one with weight 5 and one with weight 7, you can measure items of weight 5, 7, or 12. You can also measure items of weight 2, by placing 7 on one side and 5 on the side which contains the object to be weighed. If the scale balances, you know that the item has weight 2 since 5+2=7. Your goal is to measure items of weight N, where N is any whole number between 1 and 40. (a) Suppose you have six objects, with weights 1, 2, 4, 8, 16, 32. Clearly explain how you can use the balance scale to measure an item of weight N, where N is any whole number between 1 and 40. (b) Suppose you have four objects, with weights 1, 3, 9, 27. Clearly explain how you can use the balance scale to measure an item of weight N, where N is any whole number between 1 and 40. (c) Suppose you have three objects, with weights A, B, C. Determine whether there exists a set of three weights A, B, C for which you can use the balance scale to measure an item of weight N, where N is any whole number between 1 and 40. If these three weights exist, determine the values of A, B, C. If these three weights do not ex- ist, clearly explain why this problem is impossible to solve.