Cryptarithmetic Multiplication Problem

This type of problems can be solved by back-tracking technique after finding one letter (number). Approach (Method): The best metho...

This type of problems can be solved by back-tracking technique after finding one letter (number).

Approach (Method):

The best method which I follow is from bottom to top multiplication (You can do in vice-verse), but I found it was easy.
If you see the question, start from bottom to top like D*(ABC) and E*(ABC)
It is given that E * C = C this happens only with numbers of 5 & 6 let’s see how?

Explanation:

1 * 5 = 5
3 * 5 = _5 (Actually 15 but concentrate on last digit)
7 * 5 = _5 (35) I am not using 5 * 5 because it’s already assigned to C we should not repeat.
9 * 5 = _5 (45) so, from this don’t conclude C = 5 & E= (1, 3, 7, 9).
Because there is another number with these properties Ex: – number (6)
6 * 2 = _2 (12)
6 * 4 = _4 (24)
6 * 8 = _8 (48)
From this don’t conclude E = 6 & C = (2, 4, 8).
Let, note all the possible cases and try. For Example take C = 5 and E = odd number (1, 3, 7, 9)

	A	B	C
		D	E
	F	E	C
D	E	C
H	G	B	C

 Now Replace C with 5 and redraw the following table.
 And also place E as odd number (3, 7, 9) I didn’t write 1 here because it’s not  going to be 1 Let’s take first E = 3 and fill the table.

	A	B	5
		D	3
	F	3	5
D	3	5
H	G	B	5

If you see the colored portion then sum 3 + 5 = B (8) lets assign 8 to B
And redraw the table, if u put B = 8, let’s check from first




3 * 5 = 5 (carry 1)
3 * 8 = 24 + 1 = 5 (carry 2) but in the table it is showing 3 it is contradictory, then we can understand that E! = 3.
So Next I will take E = 7 and again fill the table.

	A	B	5
		D	7
	F	7	5
D	7	5
H	G	B	5

	A	2	5
		D	7
	F	7	5
D	7	5
H	G	2	5

If you see the red color portion
Sum of 7 + 5 = 2 (carry 1) so we got B = 2.
Assign B = 2 and fill the table.




Now see the row which I highlighted and think what value we can keep for D after clear Observation we can get D = 3 satisfy the given condition and put D = 3.

	A	2	5
		3	7
	F	7	5
3	7	5
H	G	2	5

	1	2	5
		3	7
	8	7	5
3	7	5
4	6	2	5

To get 375 in second row the value of A is
A = 1 so put value of A in table
If you keep A = 1 it’s very simple



F = 8, G = 6, H = 4.