Which of the following is an integer? I. $\frac{15... | Free GMAT Practice Question | GMAT Panda

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Question

Which of the following is an integer? I. \frac{15 !}{10 !} II. \frac{13 !}{7 !} III. \frac{9 !}{5 ! 6 !}

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
$Statement I: We simplify \frac{15 !}{10 !} by canceling the common factorial.$	$\frac{15 !}{10 !} = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 !}{10 !} = 15 \times 14 \times 13 \times 12 \times 11$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since the result of Statement I is a product of integers, it is an integer.$	$15 \times 14 \times 13 \times 12 \times 11 is an integer$
$Statement II: We simplify \frac{13 !}{7 !} by canceling the common factorial.$	$\frac{13 !}{7 !} = \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{7 !} = 13 \times 12 \times 11 \times 10 \times 9 \times 8$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since the result of Statement II is a product of integers, it is an integer.$	$13 \times 12 \times 11 \times 10 \times 9 \times 8 is an integer$
$Statement III: We simplify \frac{9 !}{5 ! 6 !} by canceling the 5! factor.$	$\frac{9 !}{5 ! 6 !} = \frac{9 \times 8 \times 7 \times 6 \times 5 !}{5 ! \times 6 !} = \frac{9 \times 8 \times 7 \times 6}{6 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$We factor numerator and denominator and simplify to check if the result is a whole number.$	$\frac{9 \times 8 \times 7 \times 6}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = \frac{6 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = \frac{9 \times 8 \times 7}{120} = \frac{504}{120} = 4 + \frac{24}{120} = 4.2$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since 4.2 is not a whole number, the result is not an integer.$	$4.2 is not an integer$

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Final Answer

D

Question

Which of the following is an integer? I. \frac{15 !}{10 !} II. \frac{13 !}{7 !} III. \frac{9 !}{5 ! 6 !}

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
$Statement I: We simplify \frac{15 !}{10 !} by canceling the common factorial.$	$\frac{15 !}{10 !} = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 !}{10 !} = 15 \times 14 \times 13 \times 12 \times 11$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since the result of Statement I is a product of integers, it is an integer.$	$15 \times 14 \times 13 \times 12 \times 11 is an integer$
$Statement II: We simplify \frac{13 !}{7 !} by canceling the common factorial.$	$\frac{13 !}{7 !} = \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{7 !} = 13 \times 12 \times 11 \times 10 \times 9 \times 8$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since the result of Statement II is a product of integers, it is an integer.$	$13 \times 12 \times 11 \times 10 \times 9 \times 8 is an integer$
$Statement III: We simplify \frac{9 !}{5 ! 6 !} by canceling the 5! factor.$	$\frac{9 !}{5 ! 6 !} = \frac{9 \times 8 \times 7 \times 6 \times 5 !}{5 ! \times 6 !} = \frac{9 \times 8 \times 7 \times 6}{6 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$We factor numerator and denominator and simplify to check if the result is a whole number.$	$\frac{9 \times 8 \times 7 \times 6}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = \frac{6 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = \frac{9 \times 8 \times 7}{120} = \frac{504}{120} = 4 + \frac{24}{120} = 4.2$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since 4.2 is not a whole number, the result is not an integer.$	$4.2 is not an integer$

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Final Answer

D