Problem Solving

Hi All,

We're asked for the largest value of N for which 10^N is a factor of 50! This question comes down to 'prime factorization' and requires a degree of thoroughness on your part (to make sure that you find all of the 10s in 50!)

To start, 50! = (50)(49)(48)(47)(46)(45).....(3)(2)(1), so 50! is a really big number. You do not have to calculate it though; you just have to find all of the 10s that are hidden in that product.

10 = (2)(5), so we should start by looking for all of the multiples of 5 in 50!...
5, 10, 15 and 20 each contain a "5"
25 actually contains TWO 5s (re: 5x5)
30, 35, 40 and 45 each contain a "5"
50 contains TWO 5s (re: 2x5x5)

Thus, there are 4 + 2 + 4 + 2 = TWELVE 5s

Each of those 5s can be multiplied by a 2 (and there are LOTS of 2s in 50!), so we'll end up with twelve 10s. Thus, the maximum possible value of N is 12.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

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Contact Rich at Rich.C@empowergmat.com

$$N \geqslant 0\,\,\operatorname{int} \,\,{\text{such}}\,\,{\text{that}}\,\,\,\frac{{50!}}{{{{10}^N}}} = \operatorname{int} \,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\boxed{\,N \geqslant 0\,\,\operatorname{int} \,\,{\text{such}}\,\,{\text{that}}\,\,\,\frac{{50!}}{{{5^N}}} = \operatorname{int} \,\,}$$
$$\left( * \right)\,\,{\rm{5s}}\,\,{\rm{are}}\,\,{\rm{fewer}}\,\,{\rm{than}}\,\,{\rm{2s}}\,\,$$
$$? = N\max $$
$$? = \left\lfloor {\frac{{50}}{5}} \right\rfloor + \left\lfloor {\frac{{50}}{{{5^2}}}} \right\rfloor + \underbrace {\left\lfloor {\frac{{50}}{{{5^3}}}} \right\rfloor \, + \ldots }_0 = 10 + 2 = 12\,\,$$
Full explanation:

https://www.beatthegmat.com/if-n-is-the-greatest-positive-integer-for-which-5-n-is-a-fac-t304112.html#819465


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

Regards,
Fabio.

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Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

To determine the largest value of N, we need to determine how many times 10 divides 50!. Since 10 breaks into primes of 5 and 2, and since there are there are fewer 5s in 50! than 2s, we can find the number of 5s and thus be able to determine the number of 5-and-2 pairs.

To determine the number of 5s within 50!, we can use the following shortcut in which we divide 50 by 5, then divide the quotient of 50/5 by 5 and continue this process until we no longer get a nonzero quotient.

50/5 = 10

10/5 = 2

Since 2/5 does not produce a nonzero quotient, we can stop.

The final step is to add up our quotients; that sum represents the number of factors of 5 within 50!.

Thus, there are 10 + 2 = 12 factors of 5 in 50! This also indicates that there are twelve 5-and-2 pairs, and hence there are 12 factors of 10 within 50!

Answer: C

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