$a_3$, $a_5$ and $a_0$ terms are required?

Question

We have an arithmetic sequence $a_n>0$ and it's increasing.

and we've two systems of equations:

$a_4=15$, $m+d=21$ whereas $m=lcm(a_3,a_5)$, $d=\gcd(a_3,a_5)$.

What are the values of $a_3$, $a_5$ and $a_0$

ok when i'm trying to solve it i got this :

$a_3+a_5=30$ // $a_n+a_{n+2}=2a_{n+1}$

$m*d=a_3*a_5$ // $lcm(a,b)*\gcd(a,b)=a*b$

$m+d=21$

but i get stuck here.

Answer 1

HInt: You have $d|m$, so $d$ must be a factor of $21$, and $d$ must also be a factor of $30$, which leaves only $1,3$ as possibilities.

Answer 2

Any arithmetic sequence can be written as $\{a,a+c,a+2c,a+3c,a+4c,a+5c,\ldots\}.$ Since your fourth term is $15$, plug in $a+4c = 15.$ Seeing how so many of your formulas involve multiples of three, one can easily check that $a=3$ and $c=3$ solve describe your sequence. So $\{a_n\} = \{3, 3+3, 3+6, 3+9, 3+12, 3+15, \ldots\}$ implies $a_3 = 12$ and $a_5 = 18.$