A fraction

Number Theory Level pending

Numbers x , y , z , u , v x,\,y,\,z,\,u,\,v are integers. If the fraction x v + y z v + u \dfrac{xv+y}{zv+u} can be reduced by w w , then x u − y z x u-yz is divisible by w w .

yes no cannot be determined

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1 solution

There holds x v + y = w m xv+y=wm and z v + u = w n zv+u=wn . We multiply the first equality by z z , the second by x x , afterwards subtract the first equality from the second one, and obtain x u − y z = w ( n − m ) x u-yz=w(n-m) , which means, x u − y z x u-yz is divisible by w w .

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