a.What twodigit number equals the product of units' digit by the factorial of tens' digit?
b. What if a nondecimal system (base below ten) were allowed?
(In reply to
beyond base 10, analytical solution by Daniel)
ok, I thought I had checked to make sure that the b calculated would always be greater than x and y but I see I had made a mistake in my calculation. Below is a derivation of restrictions on x,y to allow for a viable base
we need b>x and b>y
I'll start with b>x
This gives
y(x!1)/x>x
y(x!1)>x^2
y>x^2/(x!1)
now for b>y
y(x!1)/x>y
y(x!1)>yx
y(x!1x)>0
so y>0 and x!1x>0
x!>x+1
this is true for all x>2
So to summarize, it would seem that as long as we have both
x>2
and
y>x^2/(x!1)
Then we will get a valid base with b=y(x!1)/x

Posted by Daniel
on 20180601 11:35:36 