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Number Theory Level pending

This problem is the new version of my previously posted question, which received many reports because of lack of details.

If a , b , c , and d are positive integers such that

a 3 = b 2 a^{3} = b^{2}

c 3 = d 2 c^{3} = d^{2}

c − a = 5 c - a = 5

Find the value of c + a c + a .

Hint:It's not necessary to solve for b and d .


The answer is 13.

Kenneth Gravamen by Kenneth Gravamen , 23, Philippines

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1 solution

Zhaochen Xie
Sep 9, 2015

Since a^3 = b^2 a must be square itself, and same to c. using square number sequence we can know that the two squares which have an difference of 5 is 4 and 9 so the answer is 13

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