Question

10. Let a, b, c, d be integers such that ad - bc > 1. Prove that there is at least oneamong a, b, c, d which is not divisible by ad - bc.(it's a Pathfinder question)​

Answer 1

Assume all among $a,\ b,\ c,\ d$ are exactly divisible by $ad-bc.$

So let,

• $ad-bc=k$

And let,

• $a=kp$

• $b=kq$

• $c=kr$

• $d=ks$

for every $p,\ q,\ r,\ s\in\mathbb{Z}.$

We see that,

$\longrightarrow (ad-bc)^2=a^2d^2+b^2c^2-2abcd$

$\longrightarrow k^2=a^2d^2+b^2c^2-2abcd$

Substituting values of $a,\ b,\ c,\ d,$

$\longrightarrow k^2=(kp)^2(ks)^2+(kq)^2(kr)^2-2(kp)(kq)(kr)(ks)$

$\longrightarrow k^2=k^4p^2s^2+k^4q^2r^2-2k^4pqrs$

$\longrightarrow k^4p^2s^2+k^4q^2r^2-2k^4pqrs-k^2=0$

$\longrightarrow k^2(k^2p^2s^2+k^2q^2r^2-2k^2pqrs-1)=0$

$\longrightarrow k^2((kps)^2+(kqr)^2-2\cdot kps\cdot kqr-1)=0$

$\longrightarrow k^2((kps-kqr)^2-1)=0$

This implies,

$\longrightarrow k^2=0$

$\longrightarrow k=0$

$\longrightarrow ad-bc=0\quad\quad\dots(1)$

Or,

$\longrightarrow (kps-kqr)^2-1=0$

$\longrightarrow (kps-kqr)^2=1$

$\longrightarrow kps-kqr=\pm1$

$\longrightarrow k(ps-qr)=\pm1$

$\longrightarrow (ad-bc)(ps-qr)=\pm1\quad\quad\dots(2)$

Here both $ad-bc$ and $ps-qr$ are integers.

If product of two integers is equal to 1, then the two integers are either 1 or -1 each at same time.

$\longrightarrow xy=1\quad\implies\quad\left\{\begin{array}{l}x=y=1\\x=y=-1\end{array}\right.,\quad x,\ y\in\mathbb{Z}$

If product of two integers is equal to -1, then surely one among them will be 1 and the other will be -1.

$\longrightarrow xy=-1\quad\implies\quad\left\{\begin{array}{ll}x=1,&y=-1\\x=-1,&y=1\end{array}\right.,\quad x,\ y\in\mathbb{Z}$

However, from (1) and (2), we get,

$\longrightarrow ad-bc\in\{-1,\ 0,\ 1\}$

$\Longrightarrow ad-bc\leq1$

Thus we got a statement.

$\begin{minipage}{10cm}\textit{``If all among a,\ b,\ c,\ d\in\mathbb{Z} are exactly divisible by ad-bc , then ad-bc\leq1 ."}\end{minipage}$

Taking the contrapositive, we get,

$\begin{minipage}{10cm}\textit{``If ad-bc>1 for some a,\ b,\ c,\ d\in\mathbb{Z} , then there exists atleast one among a,\ b,\ c,\ d which is not exactly divisible by ad-bc ."}\end{minipage}$

Hence Proved!

Answer 2

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