Too many divisors??.....

Number Theory Level pending

Let m and n be two positive integers and $t(x)$ =the number of divisors of x .

If, $t(m)=2014$

$t(n)=2002$

$t(gcd(m,n))=11$

Then,

$t(lcm(m,n))=?$

1 solution

Fatin Farhan
Dec 28, 2014

Let $m=p_1 \times p_2^6 \times p_3^{10} \times p_4^{12}$ and $n=p_5 \times p_6^{18} \times p_7^{52}$ .

WLOG assume $p_3=p_6$ .

So, $gcd(m,n)=p_3^{10}$

$\therefore lcm(m,n)=p_1 \times p_2^6 \times p_3^{18} \times p_4^{12} \times p_5 \times p_7^{52}$ .

$\Rightarrow t(lcm(m,n))=2*7*19*13*2*53= 366548 \: _\square$