Answer:
25
Step-by-step explanation:
You want the least number of equal-length pieces that can be cut from lengths 30 cm, 36 cm, and 84 cm.
The greatest common divisor is the largest number that will divide all of these numbers evenly. A first value to check is the smallest difference between two of the numbers: 36 -30 = 6 cm.
We find that 30/6 = 5, 36/6 = 6, and 84/6 = 14, so this number divides all of them evenly.
Using a length of 6 cm, we can obtain 5 +6 +14 = 25 pieces.
25 pieces is the least number of equal lengths that can be cut from the given lengths.
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Additional comment
The GCD of three numbers is generally found by taking them two at a time. Here, we found the GCD of 30 and 36 is 6. The next step is to find the GCD of 6 and 84, which is 6.
This "find the smallest difference" is one way to make use of Euclid's algorithm for finding the GCD. That algorithm has you find the remainder from division of one number by the other. If the remainder is 0, then the divisor is the GCD. If not, replace the larger number with the remainder and repeat.
Here 36/30 = 1 r 6, and 30/6 = 5 r 0. This tells us 6 is the GCD. You will note that the remainder 6 is the difference between 36 and 30.
Euclid's algorithm works nicely when the factors may not be obvious. Here, we can also find the GCD by looking at the factors:
30 = 2·3·5
36 = 2²·3²
84 = 2²·3·7
The factors 2 and 3 are the only ones common to all numbers.
