Question

Number of points having position vector $a\hat{i}+b\hat{j}+c\hat{k},$ where $a,b,c\in \{1,2,3,4,5\}$ such that $2^a+3^b+5^c$ is divisible by $4$ is

• A. $140$
• B. $70$
• C. $100$
• D. none of these

Answer 1

Answer: (B)

$70$

$4m=2^a+3^b+5^c=2^a+{(4-1)}^b+{(4+1)}^c$

$4m=4k+{(-1)}^b+{\left(1\right)}^c$

Thus if $a=1$ and $b$ is even then $c$ is any number, and

if $a\ne 1,b=$ odd, then $c$ is any number.

Thus required number$=1\times 2\times 5\times 4\times 3\times 5=70$