[Math] Find two natural numbers, whose sum is $85$ and whose lcm is $102$.

divisibilityelementary-number-theory

I am stuck on this question,

Find two natural numbers, whose sum is $85$ and LCM is $102$.

I just broke $102$ as $17*2*3$ and saw that $85=17*2+17*3$. So numbers are $34$ and $51$.

But I need a mathematical way of solving all such kind of problems, as I just did this problem in hit and trial fashion.

Thanks

Best Answer

Here is a little twisted but different approach. Let $a$ and $b$ be the natural numbers.

Given $a+b=85\Rightarrow b=85-a$

Since $LCM(a,b)=102$

then $ab=102k\Rightarrow a(85-a) = 102k$ where k is some natural number.

Solving the above quadratic we get $$a=\frac{85\ ^+_- \sqrt{85^2-408k}}{2}$$ Since the term under the square-root should be an odd perfect square, we can easily find $k$ to be $17$.

Hence $a=51,34$.

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