Question

if e,f,g, and h represent 4 distinct, and the 4 digit number efgh x 9= hgfe, find the values of each letter.

Answer 1

We have
EFGH
X..9
----
HGFE

If a 4 digit number is multiplied by 9 and remain a 4 digit number, it has to start with a 1.

Also, we conclude that the number EFGH cannot be greater than 1111 (or else there will be carry.

So we have
1FGH
X..9
----
HGF1

9*H leaves the units digit as a 1 means that H=9.
So we have
1FG9
X..9
----
9GF1

Since the largest number is 1111, and we have a 9, one of the digits F or G must be a zero.

If G=0, F=8 and there will be carry to exceed 4 digits.

So F=0, to give:
10G9
X..9
----
9G01
The only value of digit G that gives a 0 (with carry of 8) is G=8, so 9*8+8=80

The number is therefore 1089.

Multiply by 9 and check the result.