Q55 / 90 · KVPY 2015 (SB/SX) · Mathematics

Suppose $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are positive integers such that $2^{a}+4^{b}+8^{c}=328 .$ Then $\frac{\alpha+2 b+3 x}{a b c}$ is equal to

  • A
    $\frac{1}{2}$
  • B
    $\frac{5}{8}$
  • Correct answer
    C
    $\frac{17}{24}$
  • D
    $\frac{5}{6}$

Solution

$2^{3} \times 41=2^{2}\left[1+2^{2 b-a}+2^{3 e-a}\right]$ $a=3$ $2^{3 c-a}+2^{2 b-a}=2^{3} \times 5$ $3 c-a=3$ $2 b-3 c=2$ $a=3, c=2, b=4$ $\frac{a+2 b+3 c}{a b c}=\frac{17}{24}$

Hence, the correct answer is option: 'C'.

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