2010 AMC 12B Problem 25

Since $2010=2\cdot3\cdot5\cdot67,$ write the product as $2^A3^B5^C67^D$ times a factor coprime to all four primes; then $m=\min(A,B,C,D).$ Prime $2:$ $\operatorname{pow}(n)$ is a power of $2$ only when $n=2^k.$ Since $2^{12}=4096\lt5300\lt2^{13},$ the values $k=1,\ldots,12$ contribute $A=1+2+\cdots+12=78.$ Prime $67:$ $\operatorname{pow}(n)=67$ when $67$ is the largest prime factor, i.e. $n=67j$ with $1\le j\le79$ and every prime factor of $j$ at most $67;$ excluding $j=67, 71, 73, 79$ leaves $75$ values. The one $n$ with $\operatorname{pow}(n)=67^2$ is $n=67^2\lt5300,$ adding $2.$ So $D=75+2=77.$ Prime $3:$ For $n=2^a3^b$ with $b=1,2,3,4,$ the numbers of allowable exponents $a$ are $11,10,8,7,$ respectively. These terms alone contribute $$\begin{aligned} B&\ge11+2\cdot10 \\ &\quad+3\cdot8+4\cdot7=83. \end{aligned}$$ Prime $5:$ Write $n=2^a3^c5^b.$ For $b=1,$ the counts over $c=0,\ldots,6$ total $11+9+7+6+4+3+1=41.$ For $b=2,$ the counts total $8+7+5+3+2=25,$ each contributing two factors of $5.$ Hence $C\ge41+2\cdot25=91.$ Therefore $m=\min(78,B,C,77)=77.$ Thus, the correct answer is D.