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Using its own digits (Posted on 2018-05-31) Difficulty: 3 of 5
a.What two-digit number equals the product of units' digit by the factorial of tens' digit?
b. What if a non-decimal system (base below ten) were allowed?

See The Solution Submitted by Ady TZIDON    
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re: beyond base 10, analytical solution Comment 9 of 9 |
(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!-1-x)>0
so y>0 and x!-1-x>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 2018-06-01 11:35:36
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