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In the figure above, each letter represents a distinct single digit among .0,1,2,…,9. They are positioned in such a way that A×B×C=B×G×E=D×E×F.

What digit does the letter G represent?

Dec 16, 2019

#1
+30
+2

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):

8      3

9  2  4

1      6

Answer: The value of G is 2.

Dec 17, 2019
edited by chenxander  Dec 17, 2019

#1
+30
+2

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):

8      3

9  2  4

1      6

Answer: The value of G is 2.

chenxander Dec 17, 2019
edited by chenxander  Dec 17, 2019
#2
+106519
+1

Very nice, chenxander   !!!!!

CPhill  Dec 17, 2019
#3
+30
0

Thanks! It took me about five minutes of pure guessing-and-checking to get the answer but I guess it was worth it.

chenxander  Dec 18, 2019