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**QUESTION:**

Consider the following in decimal notations (999)X(abc) = def132, determine the digits a,b,c,d,e,f.

{I know that this question has already been answered by MeritNation Expert in one of the threads... but, in that the expert didn't mention any process to solve. The expert only wrote the number. So, please do this using a process. Please don't give any link, or certified answer. }

Please find below the solution to the asked query:

We have : (999) $\times $ (abc) = def132

Here we do not need to find out d,e,f, as they're dependent. We only need to find out a,b,c.

We write given expression , As :

( 1000 - 1 )$\times $ ( abc ) = def132

abc000 - abc = def132 --- ( 1 )

Now we take L.H.S. :

abc000 - abc and sove , As :

So,

abc000 - abc = ab(c-1)(9-a)(9-b)(10-c) , Now from equation 1 we get :

ab(c-1)(9-a)(9-b)(10-c) = def132

Now we compare both side and get :

10 - c = 2 ,

**c = 8**

And

9 - b = 3 ,

**b = 6**

And

9 - a = 1 ,

**a = 8**

So,

(999) $\times $ (abc) = (999) $\times $ (868) = 867132 , Now we compare that with given equation and get :

867132 = def132 , comparing both side we get :

d = 8 , e = 6 and f = 7

Therefore,

**a = 8 , b = 6 , c = 8 , d = 8 , e = 6 and f = 7 ( Ans )**

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