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# If t is an integer, is 3^t a factor of 21!?

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Joined: 21 Oct 2013
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If t is an integer, is 3^t a factor of 21!?  [#permalink]

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21 Aug 2014, 09:10
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Difficulty:

55% (hard)

Question Stats:

61% (01:37) correct 39% (01:42) wrong based on 383 sessions

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If t is an integer, is 3^t a factor of 21!?

(1) t is the product of two distinct single-digit prime numbers that are smaller than 7.
(2) 0 < t < 9
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Joined: 02 Sep 2009
Posts: 58327
If t is an integer, is 3^t a factor of 21!?  [#permalink]

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21 Aug 2014, 14:35
3
5
If t is an integer, is 3^t a factor of 21!?

The power of 3 in factorization of 21! is 21/3 + 21/3^2 = 7 + 2 = 9. (Check here: everything-about-factorials-on-the-gmat-85592.html)

(1) t is the product of two distinct single-digit prime numbers that are smaller than 7. If t = 2*3 = 6 < 9, then 3^t will be a factor of 21! but if t = 3*5 = 15 > 9, then 3^t won't be a factor of 21!. Not sufficient.

(2) 0 < t < 9. For any nonnegative t less than or equal to 9, 3^t is a factor of 21!. Sufficient.

Check Trailing Zeros Questions and Power of a number in a factorial questions in our Special Questions Directory.
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Re: If t is an integer, is 3^t a factor of 21!?  [#permalink]

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21 Aug 2014, 10:13
21! has 3,6,9,12,15,18,21 as multiples of 3... which makes 3^9 as the maximum power of 3 which can be a factor of 21!

STAT1
Prime numbers can be 2,3 and 3,5
if its 2,3 then t= 6. So, 3^t will be a factor of 21!
if its 3,5 then t=15. So, 3^t will NOT be a factor of 21!
So, NOT Sufficient

STAT2
for all values of t from 1 to 8 3^t is a factor of 21! as 3^9 is a factor of 21!
So, SUFFICIENT

Hope it helps!
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Re: If t is an integer, is 3^t a factor of 21!?  [#permalink]

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05 Apr 2016, 18:28
goodyear2013 wrote:
If t is an integer, is 3^t a factor of 21!?

(1) t is the product of two distinct single-digit prime numbers that are smaller than 7.
(2) 0 < t < 9

21! = 1x2x3x4x5x...21
there are 9 factors of 3 here.

1. if 2x3=6 then yes
if 3x5=15 - then no.

A D are out.

2. 0<t<9
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Re: If t is an integer, is 3^t a factor of 21!?  [#permalink]

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03 Apr 2018, 09:01
Bunuel wrote:
If t is an integer, is 3^t a factor of 21!?

The power of 3 in factorization of 21! is 21/3 + 21/3^2 = 7 + 2 = 9. (Check here: http://gmatclub.com/forum/everything-ab ... 85592.html)

(1) t is the product of two distinct single-digit prime numbers that are smaller than 7. If t = 2*3 = 6 < 9, then 3^t will be a factor of 21! but if t = 3*5 = 15 > 9, then 3^t won't be a factor of 21!. Not sufficient.

(2) 0 < t < 9. For any nonnegative t less than or equal to 9, 3^t is a factor of 21!. Sufficient.

Check Trailing Zeros Questions and Power of a number in a factorial questions in our Special Questions Directory.

Thanks for another bunch of practice question on trailing zero concepts Bunuel. Always grateful for your contribution to the forum.
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Re: If t is an integer, is 3^t a factor of 21!?  [#permalink]

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09 Jun 2019, 12:43
BrushMyQuant wrote:
21! has 3,6,9,12,15,18,21 as multiples of 3... which makes 3^9 as the maximum power of 3 which can be a factor of 21!

STAT1
Prime numbers can be 2,3 and 3,5
if its 2,3 then t= 6. So, 3^t will be a factor of 21!
if its 3,5 then t=15. So, 3^t will NOT be a factor of 21!
So, NOT Sufficient

STAT2
for all values of t from 1 to 8 3^t is a factor of 21! as 3^9 is a factor of 21!
So, SUFFICIENT

Hope it helps!

if we combine both, then t=6 is also correct
Re: If t is an integer, is 3^t a factor of 21!?   [#permalink] 09 Jun 2019, 12:43
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