The number of points having position vector `a hat i + b hat j + c hatk `, where `1 leq a,b,cleq10` and `a,b,c in N`, such that `2^a + 3^b+5^c` is a multiple of 4 is (A) 1000 (B) 500 (C) 250 (D) 125

Options

A. 70
B. 140
C. 210
D. 280

Answer

A

Explanation

`because 2^(a)+3^(b)+5^(c)=2^(a)+(4-1)^(b)+(4+1)^(c)`
`=2^(a)+4k+(-1)^(b)+(1)^(c)`
`=2^(a)+4k+(-1)^(b)+1`
I. a=1,b=even, c=any number
II. `a ne 1,b=`odd, c=any number
`therefore`Required number of ways`=1xx2xx5+4xx3xx5=70`
[`because`even numbers=2,4, odd numbers=1,3,5 and any numbers=1,2,3,4,5]