Aptitude Discussion

Q. |
How many factors of $2^5 × 3^6 × 5^2$ are perfect squares? |

✖ A. |
$20$ |

✔ B. |
$24$ |

✖ C. |
$30$ |

✖ D. |
$36$ |

**Solution:**

Option(**B**) is correct

Any factor of this number should be of the form $2^a × 3^b × 5^c$.

For the factor to be a perfect square $a, b, c$ have to be even.

$a$ can take values $0, 2, 4$. $b$ can take values $0,2, 4, 6$ and $c$ can take values $0,2$.

Total number of perfect squares,

$= 3 × 4 × 2$

$= \textbf{24 }$

**AKANCHHA**

*()
*

Akansha, Its very easy.

See the number given is $2^5\times 3^6\times 5^2$, a factor is a number which divides the number fully.

So factor will have to be multiple of 2, 3 and 5 or we can say it should be of the form $2^a\times 3^b\times 5^c$.

BUT, , factor should divide the given number fully so the powers of 2, 3 and 5(i.e. a, b, c in this case) must not exceed the respective powers in number or in simpler words we have these conditions:

$a \leq 5$, $b \leq 6$ and $c \leq 2$

Now as per the questions out of all possible factors we need to select only SQUARE values for which a, b and c can take only even values.

So $a= 0,2,4$; or it can take **3** values

$b=0,2,4,6$; or it can take **4** values

$c=0,2$ or it can take **2** values

So total number of factors:

$=3\times 4\times 2$

$=\textbf{24}$

I DONT UNDERSTAND