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1. The sum of the squares of two positive integers is 100 and the difference of their squares is 28. The sum of the numbers is ?

A. 12

B. 13

C. 14

D. 15

Solution:

Let the positive integers be a and b where a > b
According to the question,

\begin{aligned} a^{2} + b^{2} = 100..... (i) \\ a^{2} - b^{2} = 28..... (ii) \end{aligned}

By adding (i) and (ii), we get :

\begin{aligned} ∴ a^{2} + b^{2} + a^{2} - b^{2} = 100 + 28 \\ \Rightarrow 2 a^{2} = 128 \\ \Rightarrow a^{2} = \frac{128}{2} \\ \Rightarrow a = \sqrt{64} \\ \Rightarrow a = 8 \end{aligned}

From equation (i)

\begin{aligned} \Rightarrow 8^{2} + b^{2} = 100 \\ \Rightarrow b^{2} = 100 - 64 \\ \Rightarrow b = \sqrt{36} \\ \Rightarrow b = 6 \\ ∴ a + b = 8 + 6 \\ = 14 \end{aligned}

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