Question

Two numbers $a$ and $b$ are selected from the set of natural number then the probability that $(a^2+b^2)$ is divisible by $5$ is

• A. $\frac{9}{25}$
• B. $\frac{7}{18}$
• C. $\frac{11}{36}$
• D. $\frac{1}{3}$

Answer 1

The correct option is A $\frac{9}{25}$

$a,b$ selected from set of natural numbers

$a^2+b^2$

Any natural no can be written as $(5q,5q\pm 1,5q\pm 2)q\in z$

only $(5q{)}^2+(5q{)}^2$ is divisible by $5$

$(5q\pm 1{)}^2+(5q\pm 2{)}^2$ is divisible byu $5$

Probability that both no are of form $5q=\frac{1}{5}\times \frac{1}{5}=\frac{1}{25}$

Probability that one no from $(5q+\pm 1)$

& second no from $(5q\pm 2)=2!\times \frac{2}{5}\times \frac{2}{5}=\frac{8}{25}$

and vice versa

Total probability $=\frac{9}{25}$

$A$ is correct