Four empty cartons with capacities of 132 litres, 78 litres, 108litres and 102 litres are to be filled with packets of sugar . What is the volume of the largest packets of sugar that can be used to fill each
carton without any leftover space or sugar packets?
To find the volume of the largest packet of sugar that can be used to fill each carton, we need to find the greatest common divisor (GCD) of the capacities of the four cartons.
Capacity of cartons: 132 litres, 78 litres, 108 litres, and 102 litres.
First, find the GCD of the first two capacities:
132 = 2 × 2 × 3 × 11 (prime factorization)
78 = 2 × 3 × 13 (prime factorization)
GCD(132, 78) = 2 × 3 = 6 litres
Next, find the GCD of the result and the third capacity:
108 = 2 × 2 × 3 × 3 × 3 (prime factorization)
GCD(6, 108) = 2 × 3 = 6 litres
Finally, find the GCD of the result and the fourth capacity:
102 = 2 × 3 × 17 (prime factorization)
GCD(6, 102) = 2 × 3 = 6 litres
Therefore, the volume of the largest packet of sugar that can be used to fill each carton without any leftover space or sugar packets is 6 litres.