The lengths of three wires were 30m, 36 m and 84m. Pieces of wire of equal length were cut from the three wires. Calculate the least number of pieces obtained.
If the pieces are even meters in length, then each piece can be 1 meter long.
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To find the least number of pieces obtained when cutting pieces of equal length from three wires, we need to determine the greatest common divisor (GCD) of the lengths of the wires.
Using the GCD, we can find the length of the wire that can be cut into equal pieces without any remainders. The GCD represents the largest length of wire that can be divided evenly into all three wires.
Let's find the GCD of the lengths 30m, 36m, and 84m:
1. Prime factorize each length:
- 30m = 2 * 3 * 5
- 36m = 2^2 * 3^2
- 84m = 2^2 * 3 * 7
2. Identify the common factors:
- The common factors of 30m, 36m, and 84m are 2 (appears twice) and 3 (appears once).
3. Calculate the GCD:
- Multiply the common factors together: GCD = 2 * 3 = 6
The GCD of 30m, 36m, and 84m is 6m.
Therefore, the least number of equal pieces that can be obtained is equal to the length of the wire divided by the GCD:
- For the 30m wire: 30m / 6m = 5 pieces
- For the 36m wire: 36m / 6m = 6 pieces
- For the 84m wire: 84m / 6m = 14 pieces
To obtain the least number of pieces, we can take the minimum count among the three wires, which is 5 pieces.
Hence, the least number of pieces obtained when cutting equal lengths from the three wires is 5.