Question

If $15!=2^{\alpha }\cdot 3^{\beta }\cdot 5^{\gamma }\cdot 7^{\delta }\cdot {11}^{\theta }\cdot {13}^{\Phi }$, then the value of expression $\alpha -\beta +\gamma -\delta +\theta -\Phi$ is

• A. $4$
• B. $6$
• C. $8$
• D. $10$

Answer 1

The correct option is B $6$

$15!=1\times 2\times 3\times 4\times 5\times 6\times 7\times 8\times 9\times 10\times 11\times 12\times 13\times 14\times 15$

$15!=2\times 3\times 2^2\times 5\times 3\times 2\times 7\times 2^3\times 3^2\times 2\times 5\times 11\times 3\times 2^2\times 13\times 2\times 7\times 3\times 5$

$15!=2^{11}\times 3^6\times 5^3\times 7^2\times 11\times 13$

$⟹\alpha =11$

$⟹\beta =6$

$⟹\gamma =3$

$⟹\delta =2$

$⟹\theta =1$

$⟹\Phi =1$

$\alpha -\beta +\gamma -\delta +\theta -\Phi =11-6+3-2+1-1=6$