Find the numbers that each letter stands for. EFGHx4=HGFE

E=
F=
G=
H=

E must be 1 or 2 so than 4xEFGH can be a four-digit number. E must be 2 because multiplying EFGH by 4 would required that E be an even number.

H must therefore be 8 since E is 2.

Thus EFGH is 2--8 and HGFE is 8--2.
In order for HGFE to begin with 8, F must be 1 or 2. Let's try 1, since we have a 2 already.

Now we have 21_8 = 8_12

With only ten digits to try, it is easy to see that the numbers are 2178 and 8712

To find the values of the letters E, F, G, and H, we need to solve the given equation EFGHx4=HGFE.

Let's break down the equation step by step:

1. Rearrange the equation to isolate the product term:
EFGHx4 = HGFE

2. Divide both sides of the equation by 4 to isolate the product term on the left side:
EFGH = HGFE/4

3. Since the left side is a four-digit number, and the right side HGFE is also a four-digit number divided by 4, we know that G must equal 1, as no other value would make the product a four-digit number.

4. Substituting G = 1 into the equation, we get:
EF1H = H1FE/4

5. Since the product of a four-digit number divided by 4 still results in a four-digit number, we know that H must be an even number. The only even digit remaining is 2.

6. Substituting H = 2, we get:
EF12 = 21FE/4

7. Now, we can try different possible values for E, F, and 1 to find a solution:
Let's try E = 3, F = 6, 1 = 7.

Substituting these values, we get:
3612 = 2176/4

Simplifying the right side, we find:
3612 = 544

8. Since the equation is not true for E = 3, F = 6, and 1 = 7, we need to try a different set of values.

Let's try E = 8, F = 9, 1 = 5.

Substituting these values, we get:
8912 = 2195/4

Simplifying the right side, we find:
8912 = 5487.5

9. Again, the equation is not true for E = 8, F = 9, and 1 = 5.

Unfortunately, it seems that there are no values for the letters E, F, G, and H that satisfy the given equation EFGHx4 = HGFE.

To find the numbers that each letter stands for in the equation EFGHx4 = HGFE, you can start by examining the pattern.

Let's break down the equation step by step:

EFGHx4 = HGFE

First, we know that the letter "E" appears on both sides of the equation. This tells us that "E" represents the same number on both sides.

Next, we see that "EFGHx4" is multiplied by 4. Multiplying a number by 4 means that the value is multiplied by 4, so we can write this as 4EFGH.

Therefore, the equation becomes:

4EFGH = HGFE

Now, let's try to find the values for each letter individually.

Looking at the equation, we notice that the letter "H" is the only letter that appears on both sides without any operation applied to it. This means that "H" must represent a value that is the same on both sides. Consequently, "H" must be zero.

Now our equation becomes:

4EFG0 = G0FE

Now focus on the remaining letters: E, F, and G.

Since our equation involves a multiplication (4EFG0), we can deduce that G must be 1. If G were any other number, multiplying it by 4 would produce a value that does not end in a zero. Therefore, G must be 1.

Our equation now looks like this:

4EF10 = 010E

We are left with the letters E and F.

Looking at the equation, we notice that the left-hand side is a number followed by EF10. Meanwhile, the right-hand side starts with three zeros followed by E. This suggests that the right-hand side must be a larger number than the left-hand side, as it contains extra zeros before the E.

To achieve this, we can deduce that E must be 4, which is greater than zero.

Finally, we can determine that the remaining letter, F, must be 2. As there are only four unique digits in total (0, 1, 2, and 4), and we have already assigned the other three to different letters, F must be 2.

Hence, the numbers that each letter represents in the equation EFGHx4 = HGFE are:

E = 4
F = 2
G = 1
H = 0