I just need to chcek my solution
Find all pairs of natural numbers a, b that fit to an equation
$a+b+gcf(a,b)+lcm(a,b)=50$,
gcf(a,b) is the greatest common factor and lcm (a,b) is the least common multiple of a and b.
We know that a.b=lcm(a,b).gcf(a,b) So I substitue it to the equation to get rid of lcm(a,b)
$a+b+gcf(a,b)+\frac{a.b}{gcf(a,b)}=50$
i know, that numbers a and b can be rewrite with gcf like this $a=gcf.x, b=gcf.y $ when x, y must be prime numbers or their multiples as well as they cannot have any common dividend with gcf
$gcf.x+gcf.y+gcf+\frac{gcf.x.gcf.y}{gcf}=50$
$x+y+1+x.y=\frac{50}{gcf}$
$(x+1)(y+1)=\frac{50}{gcf}$
that means that gcf/50 and gcf={1,2,5,10,25,50} and I look for two numbers the product of which is $\frac{50}{gcf}$
I begin to substitute and I find out these solutions $(a,b)=(1,24),(24,1),(4,9),(9,4),(5,20),(20,5)$
is my method right?? and did I manage to find all results ??
Thanks
Your method is completely correct and you have all the results because at each step the solutions correspond to all the solutions of the previous step. It is common usage in mathematics to use gcd (greatest common divisor) instead of gcf, and use divisor instead of divident.