Find Sixth-Digit Number

[greatwhiteshark]

In the following equation, each letter represents a distinct digit.
7(ABCXYZ)=6(XYZABC) A and X are not zero. Find the 6-digit number ABCXYZ.

 

[Denis]

In the following equation, each letter represents a distinct digit.
7(ABCXYZ)=6(XYZABC) A and X are not zero. Find the 6-digit number ABCXYZ.

Hmmm....
A = [5993(100X + 10Y + Z) - 6994(10B + C)] / 699400
Hmmm....

 

[soroban]

Hello, Janet!

In the following equation, each letter represents a distinct digit.
. . . 7(ABCXYZ) .= .6(XYZABC)
A and X are not zero. Find the 6-digit number ABCXYZ.

Let P equal the three-digit number ABC.
Let Q equal the three-digit number XYZ.

Then: . "ABCXYZ" .= .1000P + Q
. and: . "XYZABC" .= .1000Q + P

The equation becomes: . 7(1000P + Q) . = . 6(1000Q + O)

. . . .which simplfies to: . . . . . . . 6994P . = . 5993Q

. . . . . . and factors to: . . . . 2·13·269·P . = . 13·461·Q

. . . . . . . . .and hence: . . . . . . . . .538P . = . 461Q


Therefore: . P = 461, .Q = 538 . and . "ABCXYZ" . = . 461,538

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. . . . . . . . . . . . . . .Check

7(461,538) . = . 3,230,766 . = . 6(538,461)