Question

Product of all the even divisors of $N=1000$, is

• A. $32\cdot 10^2$
• B. $64\cdot 2^{14}$
• C. $64\cdot 10^{18}$
• D. $128\cdot 10^6$

Answer 1

Answer: (C)

$1000=2^3\cdot 5^3$
Number of divisors of $N$ which are $\left(2^0+2^1+2^2+2^3\right)\left(5^0+5^1+5^2+5^3\right)=16$ Product of all the 16 divisors is
$=\left(5^6\right)\left(5^6\right)\left(5^6\right)\left(5^6\right)\left(2^6\right)\left(2^6\right)\left(2^6\right)\left(2^6\right)$
$=2^{24}\cdot 5^{24}$
Now odd divisors are $1,5,25,125$
and product of odd deivisors $=5^6$
$\therefore$ Product of all the even divisors $=2^{24}\cdot 5^{18}=2^{18}\cdot 5^{18}\cdot 2^6=\left(10^{18}\right)64$
$\text{ Alternatively: Prodcut of all the even divisors }=\left(2^4\cdot 5^6\right)\left(2^8\cdot 5^6\right)\left(2^{12}\cdot 5^6\right)\text{ (think!) }$
$=2^{24}\cdot 5^{18}=64\cdot 10^{18}$