Hi,
If this was to be done logically, this is how it would be like
Lets See how two numbers can be multiplied
XY
*YX
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\(\hspace{3cm}X^2 \hspace{.3cm} XY\)
\(\hspace{1.3cm} XY \hspace{1cm} Y^2\hspace{.5cm}\) 0
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\(\hspace{1.3cm} X \hspace{1cm} Z \hspace{.7cm}\) X
Now here we are given certain Conditions , that X and Y are distinct integers and product of XY < 10
The Units Place of product is obtained when we do the following : X*Y +0 = X what can we infer . Using the identity that 1*A = A we can say that Y is definitely 1. And 1*X =X
The Hundreds Place of product is obtained when we do the following : XY + any carry forward from ten's place But . Its told to us that its X . Since we have seen that Y is 1 XY is =X if there is no carry forward from Ten's Place.
The Tens Place of product is obtained when we do the following : \(X^{2}\) +\(Y^{2}\) = Z
But here we Know that Y =1 So \(X^{2}\) +1 gives us the value of Z. Also Since tens Place is a number <10, because any double digit number would lead to carry forward to hundreds place the only integer whose square is less than 10 is 2,3,
But \(2^{2}\)= 4 , \(3^{3}\)=9, If it were 3 then \(3^{2}\) would be 9 and \(Y^{2}\) =1 which would be 10, this would provide carry to hundreds digit. So X is not 3 , hence X is 2.
So we have X as 2 and Y as 1
which is 21
Probus
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Probus
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