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You can use Euclid's algorithm to calculate the gcf of two of the numbers - then use Euclid's algorithm again with the result and the third number.Or you can factor all the numbers into prime factors, and check which prime factors occur in all three numbers.
Yes. For example, the average of 5, 7, and 9 is 7.
The average of these three numbers is 10.666
The average of these three numbers is 12.
It average of 2 numbers is 9 then take 9 away from 12, so average of 3 numbers is now 9, with 3 'left over.' The 3 which is 'left over' divides among the three numbers by adding 1 to each of them, raising the new average to 10.