The number of positive integer solutions of a+b+c=60, where a is a factor of b and c, is

Options

A. 184
B. 200
C. 144
D. 270

Answer

C

Explanation

`because` a is a factor of b and `cimpliesa` divides 60
`thereforea=1,2,3,4,5,6,10,12,15,30" "[because a ne60]`
and b=ma, c=na, when m, `n ne 1`
`because a+b+c=60`
`impliesa+ma+na=60 impliesm+n=((60)/(a)-1)`
`therefore`Number of solutions`=.^((60)/(a)-1-1)C_(2-1)=((60)/(a)-1)`
Hence, total number of solutions for all values of a
`=58+28+18+13+10+8+4+3+2+0=144`