MathDB

5

Part of 2012 BMT Spring

Problems(2)

Spring Round (2012) #5

Source:

12/3/2012
Let a↑↑b=aaa…aa{ a\uparrow\uparrow b = {{{{{a^{a}}^a}^{\dots}}}^{a}}^{a}} , where there are b b a's in total. That is a↑↑b a\uparrow\uparrow b is given by the recurrence a↑↑b={ab=1aa↑↑(b−1)b≥2 a\uparrow\uparrow b = \begin{cases} a & b=1\\ a^{a\uparrow\uparrow (b-1)} & b\ge2\end{cases} What is the remainder of 3↑↑(3↑↑(3↑↑3)) 3\uparrow\uparrow( 3\uparrow\uparrow ( 3\uparrow\uparrow 3)) when divided by 60 60 ?
2012 BMT Team 5

Source:

1/5/2022
Let p>1p > 1 be relatively prime to 1010. Let nn be any positive number andd d be the last digit of nn. Define f(n)=⌊n10⌋+d⋅mf(n) = \lfloor \frac{n}{10} \rfloor + d \cdot m. Then, we can call mm a divisibility multiplier for pp, if f(n)f(n) is divisible by pp if and only if nn is divisible by pp. Find a divisibility multiplier for 20132013.
floor functionnumber theory