MHB Perfect Cube & Square: 5-Digit Number

[kaliprasad]

Find 5-digit number whose half is a perfect cube and one-third is a perfect square

 

[Greg]

Find 5-digit number whose half is a perfect cube and one-third is a perfect square

Spoiler

$$2^{10}\cdot3^3=27648$$

 

[kaliprasad]

Spoiler

$$2^{10}\cdot3^3=27648$$

right ans . here is my method

Spoiler

the number is $2n^3a^6= 3m^2a^6$ ( we can choose m, n to be as low as possible)
now $n^3$ and $m^2$ shall be of the form $2^a3^b$ say $n = 2^x3^y$ and $m = 2^p3^q$

so we get $2^{3x+1}3^{3y} = 2^{2p} 3^{2q+1}$
comparing exponents on both sides $3x + 1 = 2p$ givinng $x = 1 , p = 2$
and $3y = 2q + 1$ giving $y = q = 1$
so $n = 2 * (2 * 3)^3 = 432$
$\frac{10000}{432} = 23$ and $\frac{100000}{432} = 231$ and 64 is the only $6^{th}$ power between 23 and 231 is $2^6 = 64$
hence $ans = 64n = 432 * 64 = 27648$