Question

There are no distinct whole number a, b and c such that a ÷ (b ÷ c) = (a ÷ b) ÷ c

Answer 1

Answer:

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Step-by-step explanation:

Suppose a+b=c+d and a3+b3=c3+d3.

a+b=c+d

(a+b)3=(c+d)3

a3+b3+3ab(a+b)=c3+d3+3cd(c+d)

3ab(a+b)=3cd(c+d)

ab=cd

Let a+b=c+d=m and ab=cd=n

a and b are the roots of the quadratic equation

x2−mx+n=0

by Vieta's relations because a+b=m and ab=n. But c and d are also roots of the equation for similar reasons. But a quadratic equation can have at most two distinct roots.

Hence, a=c or a=d, so a,b,c,d are not distinct.

Mark a+b=x then b=x−a and d=x−c. Notice that x≠0. No we have:

a3+(x−a)3=c3+(x−c)3

and thus

−3x2a+3xa2=−3x2c+3xc2

so

xa−a2=xc−c2⟹x(a−c)=(a−c)(a+c)⟹x=a+c⟹a+c=a+b⟹c=b

A contradiction.