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Question

xa−rxyr−pxzp−q=1.

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Solution

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Given,

arp−1=x

arq−1=y

arr−1=z

⇒(arp−1)q−r=xq−r

⇒(arq−1)r−p=yr−p

⇒(arr−1)p−q=zp−q

xq−ryr−pzp−q=(arp−1)q−r(arq−1)r−p(arr−1)p−q

xq−rxyr−pxzp−q=a(p+q+r−p−q−r)r(pq+qr+rp+p+q+r−qp−rq−pr−p−q−r)

⇒xq−rxyr−pxzp−q=a0r0

∴xq−rxyr−pxzp−q=1

Hence proved.

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