Which of the following is an integer? I. 12!/6! II. 12!/8! III....

Question

Which of the following is an integer? I. 12!/6! II. 12!/8! III. 12!/7!5!

GMAT Panda · Accepted Answer

E. I, II, and III are all integers. Statement I: We simplify 12!/6! by canceling the common factorial. 12!/6! = 1211109876!6! = 121110987 Since the result of Statement I is a product of integers, it is an integer. 121110987 is an integer Statement II: We simplify 12!/8! by canceling the common factorial. 12!/8! = 12111098!8! = 1211109 Since the result of Statement II is a product of integers, it is an integer. 1211109 is an integer Statement III: We simplify 12!/7!5! by canceling 7! and then checking divisibility by 5!. 12!/7!5! = 121110987!7!5! = 12111098/5! We factor numerator and denominator to cancel common factors 12111098/5! = 12111098/54321. Cancel 12/4=3, 10/5=2, 9/3=3, 8/2=4, so we get 311234 which is an integer

Explanation	Calculations	Help
$Statement I: We simplify \frac{12 !}{6 !} by canceling the common factorial.$	$\frac{12 !}{6 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 !}{6 !} = 12 \times 11 \times 10 \times 9 \times 8 \times 7$
$Since the result of Statement I is a product of integers, it is an integer.$	$12 \times 11 \times 10 \times 9 \times 8 \times 7 is an integer$
$Statement II: We simplify \frac{12 !}{8 !} by canceling the common factorial.$	$\frac{12 !}{8 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 !}{8 !} = 12 \times 11 \times 10 \times 9$
$Since the result of Statement II is a product of integers, it is an integer.$	$12 \times 11 \times 10 \times 9 is an integer$
$Statement III: We simplify \frac{12 !}{7 ! 5 !} by canceling 7! and then checking divisibility by 5! .$	$\frac{12 !}{7 ! 5 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{7 ! \times 5 !} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 !}$
$We factor numerator and denominator to cancel common factors$	$\frac{12 \times 11 \times 10 \times 9 \times 8}{5 !} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} . Cancel 12/4 = 3, 10/5 = 2, 9/3 = 3, 8/2 = 4, so we get 3 \times 11 \times 2 \times 3 \times 4 which is an integer$

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