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Since the three products are the same, it is impossible for one product to have a factor that another does not.
Therefore, 7 and 5 don't work because they are coprime (don't share any factors) with every other number on the list.
Of course, 0 doesn't work because then one or two products will be zero while the other is a positive number.
The remaining numbers, 1, 2, 3, 4, 6, 8, and 9 , are the possible digits.
Notice that every number only has factors of 2 and 3. So, separate the numbers into two groups depending on their factors:
factors of 2: (2,4,6,8)
factors of three: (3,6,9)
There are a total of seven factors of 2 and four factors of 3.
This is somewhat hard to explain, but this means that you must put a factor of 2 as G because otherwise one product will always have one more factor of two than another. Since G belongs to only one product, putting a factor of two as G allows the remaining six factors of 2 to be split equally between AxBxC and DxExF.
If you aren't able to completely follow along below, it's fine, as I am just guessing-and-checking. Just scroll down some more.
Try setting 2 as G. Since 8 has three factors and 2 combined with 4 has three factors, set 8 as A and D,F as 4 and 6 respectively.
You will find that this doesn't work. Instead of putting a 2 on the side, put a six as F. Then, set B as 9 and G as 3. This will balance out the factors of three evenly.
Finally, put one as C and two as G.
My final configuration is (there are many, put they all have 2 as G):
9 2 4
Answer: The value of G is 2.
The strategy here is to find the boundary of the graph and then figure out what the actual graph is. First, plot some points for |x| + |y| = 5 on the Cartersian plane; (5,0), (4,1), (3,2), and combinations of their opposites work as well (ex. -4,1).
The graph of |x| + |y| = 5 becomes a diamond centered at the origin with perpendicular diagonals of length 10 on the x and y-axis.
If you think about it, |x| + |y| < 5 must be all of the area within the diamond.
At this point, you can count the number of lattice points by drawing a diagram. Be careful, as the inequality sign is less than, not less than or equal to, so do not count the lattice points on the diamond itself.
By counting with symmetry, you get
6*4 + 4*4 + 1 = 41.
The 6*4 is for the points between the axes, the 4*4 is the points on the axes (excluding the origin), and the 1 is for the origin.