Shane has 72 inches of copper wire and 42 inches of steel wire. What are the largest pieces he can cut these wires into so that each piece is the same length and all of the wire is used?
To find the largest common length that the wires can be cut into, we need to find the greatest common divisor (GCD) of 72 and 42. This can be done through prime factorization:
72 = 2^3 * 3^2
42 = 2 * 3 * 7
To find the GCD, we take the lowest exponent of each common prime factor:
GCD = 2^1 * 3^1 = 2 * 3 = 6
Therefore, the largest common length that the wires can be cut into is 6 inches.
To use up all the wire, we need to find how many 6-inch pieces can be cut from each wire:
72 inches of copper wire ÷ 6 inches = 12 pieces
42 inches of steel wire ÷ 6 inches = 7 pieces
Thus, the largest pieces Shane can cut the wires into so that each piece is the same length and all of the wire is used is 6 inches long.