Question

Number of ordered triplets $(p, q, r)$ where p, q, r $\in N$ lying in $[1,100]$, such that $\left(2^p+3^q+5^r\right)$ is divisible by 4 is $\lambda \cdot 10^5$ where $\lambda$ is equal to

• A. 50
• B. $\frac{1}{2}$
• C. 5
• D. $\frac{1}{20}$

Answer 1

Answer: (C)

$2^{ p }+(4-1)^{ q }+(4+1)^{ r } $
$\therefore\left(2^{ p }+(-1)^{ q }+(1)^{ r }\right) \text { must be divisible by } 4 $
$p =1 \text { and } q =\text { even } \Rightarrow 1 \times 50 \times 100$
$p >1 \text { and } q =\text { odd } \Rightarrow 99 \times 50 \times 100 $
$\therefore N =100 \times 50 \times 100=5 \times 10^5$