How about 4, 5, 6?

Number Theory Level 3

$\sqrt { ab+56 } -\sqrt { ab-56 } =4$

If all the $n$ pairs of positive integers $(a,b)$ that satisfy the equation above are

$(a_1 ,b_1) , (a_2, b_2) , \ldots , (a_n, b_n) ,$

submit your answer as $\displaystyle \sum _{ i=1 }^{ n }{ a_{ i }b_{ i } }$ .

The answer is 2400.

1 solution

Rishabh Jain
Jul 25, 2016

Relevant wiki: Number of Factors

$\sqrt { ab+56 } =4+\sqrt { ab-56 }$

Squaring we get:-

$\cancel{ab}+56=16+\cancel{ab}-56+8\sqrt{ab-56}$

$\implies 12=\sqrt{ab-56}$

Squaring again we get $ab=200$ which on direct substitution satisfies the original equation.

$ab=200=2^3\cdot 5^2$

So we need to find the ways to express $200$ as a product of two integers in order so that we can find $(a,b)$ . Note that this is equal to number of factors of $200$ i.e $(3+1)(2+1)=12$ . Hence:

$\sum_{i=1}^n a_ib_i =\sum_{i=1}^{12}(200)=200\times 12=\boxed{2400}$

As same you! Great solution.

Paola Ramírez - 4 years, 10 months ago

That's great... :-) and Thanks

Rishabh Jain - 4 years, 10 months ago

But doesn't the question state that a,b are positive integers??????

Sahil Nare - 4 years, 10 months ago

How about 4, 5, 6?

The answer is 2400.

1 solution

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