Problem Solving

[Math Revolution GMAT math practice question]

What is the smallest integer n such that 2^8 is a factor of n!?

A. 8
B. 10
C. 12
D. 14
E. 16

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

What is the smallest integer n such that 2^8 is a factor of n!?

A. 8
B. 10
C. 12
D. 14
E. 16

$$?\,\,\,:\,\,\,n \ge 0\,\,{\rm{int}}\,\,\,{\rm{min}}\,\,{\rm{such}}\,\,{\rm{that}}\,\,\,{{n!} \over {{2^8}}} = {\mathop{\rm int}} $$
$${{n!} \over {{2^8}}} = {\mathop{\rm int}} \,\,\,\,\, \Leftrightarrow \,\,\,\,n!\,\,{\rm{has}}\,\,\left( {{\rm{at}}\,\,{\rm{least}}} \right)\,\,{\rm{eight}}\,\,{\rm{2s}}$$
$$n = 8\,\,\,\,\, \Rightarrow \,\,\,\,\,8! = \left( {{2^3}} \right) \cdot 7 \cdot \left( {2 \cdot 3} \right) \cdot 5 \cdot \left( {{2^2}} \right) \cdot 3 \cdot 2\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{seven}}\,\,2{\rm{s}}$$
$${\rm{Conclusion:}}\,\,{\rm{10}}\,\,{\rm{is}}\,\,{\rm{the}}\,\,{\rm{answer}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {\rm{B}} \right)$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

n=a number from 0 to infinity i.e
$$0,1,2,3,4.................n$$
$$n!=The\ factorial\ of\ a\ number\ from\ 0\ to\ \inf inity$$
$$i.e\ 0!,1!,2!,3!,4!.......................n!$$
such that 2^8 is a factor of n!
when n=8
n!=8!
$$8!=8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=720$$
$$Factors\ of\ 720=2\cdot2\cdot2\cdot2\cdot3\cdot3\cdot5=2^4\cdot3^2\cdot5$$
$$;\ 2^8is\ not\ a\ factor\ of\ \ 8!$$
$$10!=10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=3,628,800$$
$$factors\ of\ 3,628,800\ =2^8\cdot3^4\cdot5^2\cdot7\ \ $$
$$2^{8\ }is\ a\ factor\ of\ \ 10!$$
$$12!,\ 14!\ and\ 16!\ are\ greater\ than\ 10!\ hence\ 10\ is\ the\ smallest\ integer\ such\ that\ 2^8\ is\ a\ factor\ of\ 10!$$
$$answer\ =\ Option\ B$$

=>
Since 2 = 2^1, 4 = 2^2, 6 = 2^1*3, 8 = 2^3, 10=2^1*5, the prime factorization of 10! = 1*2*3*...*10 has the form 28*m for some integer m, where m and 2 are relatively prime. Note that 9! = 2^7*k for some integer k, where k and 2 are relatively prime.

Therefore, the answer is B.
Answer: B