(4-digit number)*(6-digit number) equals a factorial

[K Sengupta]

Substitute each of the letters by a different decimal digit from 0 to 9 to satisfy this cryptarithmetic equation:

(ABCD)*(EFEGBH) = (EC)!

Note: None of A and E can be zero.

 

[Caracrist]

4725 * 101376 = 12!

 

[Architeuthis Dux]

First, let's break down the equation. We have a 4-digit number multiplied by a 6-digit number, resulting in a factorial. This means that the product of the two numbers must be a very large number, and the factorial must be equal to this product.

To solve this cryptarithmetic equation, we need to find the values for each letter (A, B, C, D, E, G, H) that satisfy the equation.

First, we know that A and E cannot be zero, as this would result in a 3-digit number when multiplied by a 4 or 6-digit number.

Next, we can start by looking at the factorial on the right side of the equation. Since the factorial is equal to the product of the two numbers, it must be a very large number. The smallest factorial that has 6 digits is 720 (6!), and the largest factorial with 6 digits is 362880 (9!).

Based on this, we can determine that E must be either 7, 8, or 9.

Now, let's look at the 4-digit number on the left side of the equation. Since A cannot be zero, it must be either 1, 2, 3, 4, 5, 6, 7, 8, or 9. This means that B, C, and D can be any number from 0 to 9.

Next, we can look at the 6-digit number on the left side of the equation. Since E is already taken and cannot be zero, G and H must be any number from 0 to 9. This means that F can be any number from 1 to 9.

Now, we have a range of numbers for each letter:

A: 1-9
B, C, D: 0-9
E: 7-9
F: 1-9
G, H: 0-9

To find a specific solution, we can start by assigning values to the letters with the most restrictions. For example, we can choose A=1, E=7, and F=1. This would give us the equation:

(1BCD)*(7G1BHG) = (7C)!

We can then start trying different combinations for the remaining letters to see if they satisfy the equation.

After some trial