Which of the following is an integer? I. $\frac{14... | Free GMAT Practice Question | GMAT Panda

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Question

Which of the following is an integer? I. \frac{14 !}{7 !} II. \frac{14 !}{9 ! \times 2 !} III. \frac{14 !}{16 !}

Answer Choices

A. $I only$
B. $II only$
C. $III only$
D. $I and II only$
E. $I, II, and III$

Steps

Explanation	Calculations	Help
$Statement I: We simplify the fraction by canceling out the common factor.$	$\frac{14 !}{7 !} = \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{7 !} = 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8$	Theory & Tactics Method Card CALC2-C Click to view full details
$We observe that the product in the previous step involves only integers, so the value is an integer.$	$14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 is an integer$
$Statement II: We cancel the common 9! factor and then simplify the remaining division by 2! .$	$\frac{14 !}{9 ! \times 2 !} = \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9 !}{9 ! \times 2 !} = \frac{14 \times 13 \times 12 \times 11 \times 10}{2 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$We simplify by canceling a factor of 2 between numerator and denominator.$	$\frac{14 \times 13 \times 12 \times 11 \times 10}{2 \times 1} = \frac{14 \times 13 \times 12 \times 11 \times 10}{2 \times 1} = 7 \times 13 \times 12 \times 11 \times 10$	Theory & Tactics Method Card CALC2-C Click to view full details
$The result is a product of integers, so it is an integer.$	$7 \times 13 \times 12 \times 11 \times 10 is an integer$
$Statement III: We express in terms of and simplify the fraction.$	$\frac{14 !}{16 !} = \frac{14 !}{16 \times 15 \times 14 !} = \frac{1}{16 \times 15} = \frac{1}{240}$	Theory & Tactics Method Card CALC2-C Click to view full details
$The value \frac{1}{240} is not a whole number, so it is not an integer.$	$\frac{1}{240} is not an integer$

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Final Answer

D

Question

Which of the following is an integer? I. \frac{14 !}{7 !} II. \frac{14 !}{9 ! \times 2 !} III. \frac{14 !}{16 !}

Answer Choices

A. $I only$
B. $II only$
C. $III only$
D. $I and II only$
E. $I, II, and III$

Steps

Explanation	Calculations	Help
$Statement I: We simplify the fraction by canceling out the common factor.$	$\frac{14 !}{7 !} = \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{7 !} = 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8$	Theory & Tactics Method Card CALC2-C Click to view full details
$We observe that the product in the previous step involves only integers, so the value is an integer.$	$14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 is an integer$
$Statement II: We cancel the common 9! factor and then simplify the remaining division by 2! .$	$\frac{14 !}{9 ! \times 2 !} = \frac{14 \times 13 \times 12 \times 11 \times 10 \times 9 !}{9 ! \times 2 !} = \frac{14 \times 13 \times 12 \times 11 \times 10}{2 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$We simplify by canceling a factor of 2 between numerator and denominator.$	$\frac{14 \times 13 \times 12 \times 11 \times 10}{2 \times 1} = \frac{14 \times 13 \times 12 \times 11 \times 10}{2 \times 1} = 7 \times 13 \times 12 \times 11 \times 10$	Theory & Tactics Method Card CALC2-C Click to view full details
$The result is a product of integers, so it is an integer.$	$7 \times 13 \times 12 \times 11 \times 10 is an integer$
$Statement III: We express in terms of and simplify the fraction.$	$\frac{14 !}{16 !} = \frac{14 !}{16 \times 15 \times 14 !} = \frac{1}{16 \times 15} = \frac{1}{240}$	Theory & Tactics Method Card CALC2-C Click to view full details
$The value \frac{1}{240} is not a whole number, so it is not an integer.$	$\frac{1}{240} is not an integer$

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Final Answer

D