Assume-that-a-b-c-and-d-are-positive-integers-such-that-a-5-b-4-c-3-d-2-and-c-a-19-Determine-d-b-

Question Number 19239 by Tinkutara last updated on 07/Aug/17

Assume that a, b, c and d are positive

integers such that a^{5} = b^{4}, c^{3} = d^{2} and

c - a = 19 . Determine d - b .

Answered by ajfour last updated on 07/Aug/17

let a^{5} = b^{4} = n^{20}

\Rightarrow a = n^{4} and b = n^{5} \dots . (i)

let c^{3} = d^{2} = m^{6}

\Rightarrow c = m^{2} and d = m^{3} \dots (ii)

Since c - a = 19

m^{2} - n^{4} = 19

or (m - n^{2}) (m + n^{2}) = 19

\Rightarrow m - n^{2} = 1 and m + n^{2} = 19

\Rightarrow 2 m = 20 and {2 n}^{2} = 18

m = 10; n = 3

So from (i) and (ii)

a = n^{4} = 81

b = n^{5} = 243

c = m^{2} = 100

d = m^{3} = 1000 .

d - b = 757 .

Commented by Tinkutara last updated on 08/Aug/17

Thank you very much Sir!

Commented by malwaan last updated on 08/Aug/17

can you solve 19204 please ?

Commented by ajfour last updated on 08/Aug/17

I know not .

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