Question

If a, b, c and d are natural numbers such that a = bc, b = cd, c = da and d = ab
Which of the following is true for the (a+b) (b+c) (c+d) (d+a)

a) (a+b+c+d)^2
b) (a+d)^2 + (b+c)^2
c) (a+b)^2 + (c+d)^2
d) (a+c)^2 + (b+d)^2

Answer 1

Answer:

Given, a = bc , b = cd , c = da , d = ab

multiplying all, we get

abcd = bc*cd*da*ab

=> abcd = abcd * abcd

=> abcd = 1

Now,

a/b = bc/cd

=> a/b =b/d

=> d=ab

=> a*ab = b2

=> a2 = b

By substituting equations, we get

d=a3

c=a4

b=a2

a=a

multiplying these all

abcd = a10

as we have earlier

abcd=1

1=a10

=> a=1

=> b=1

=> c=1

=> d=1

Now, (a+b)*(b+c)*(c+d)*(d+a) = (1+1)*(1+1)*(1+1)*(1+1) = 16

From the option,

(a+b+c+d)2 = (1+1+1+1)2 = 16