To find the shortest length of string that can be cut into lengths of 6m, 8m, or 10m, we need to find the least common multiple (LCM) of these three numbers.
One way to find the LCM is to list the multiples of each number until we find the smallest multiple that is common to all three lists.
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, ...
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, ...
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, ...
From these lists, we can see that the smallest multiple that is common to all three is 120.
Therefore, the shortest length of string that can be cut into lengths of 6m, 8m, or 10m is 120m.
We can cut this length into 20 pieces of 6m, 15 pieces of 8m, or 12 pieces of 10m.