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Number Theory Level 2

Assume that $a,b,c,d$ are positive integers, and $\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{3}{4}$ , $\sqrt{a^2+c^2}-\sqrt{b^2+d^2}=15$ .

Find $ac+bd-ad-bc$ .

101 126 81 36 72 108

1 solution

Parth Sankhe
Oct 21, 2018

$a=¾c, b=¾d$

Putting these values in the second equation,

$c-d=12$

The required expression can be factorised in the form $(a-b)(c-d)=¾(c-d)^2=¾\cdot 144=108$ .