Which of the following is an integer? I. $\frac{12... | Free GMAT Practice Question | GMAT Panda

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Question

Which of the following is an integer? I. \frac{12 !}{6 !} II. \frac{12 !}{8 !} III. \frac{12 !}{7 ! 5 !}

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
$Statement I: We simplify \frac{12 !}{6 !} by canceling the common factorial.$	$\frac{12 !}{6 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 !}{6 !} = 12 \times 11 \times 10 \times 9 \times 8 \times 7$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since the result of Statement I is a product of integers, it is an integer.$	$12 \times 11 \times 10 \times 9 \times 8 \times 7 is an integer$	Theory & Tactics Method Card NT1 Click to view full details
$Statement II: We simplify \frac{12 !}{8 !} by canceling the common factorial.$	$\frac{12 !}{8 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 !}{8 !} = 12 \times 11 \times 10 \times 9$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since the result of Statement II is a product of integers, it is an integer.$	$12 \times 11 \times 10 \times 9 is an integer$	Theory & Tactics Method Card NT1 Click to view full details
$Statement III: We simplify \frac{12 !}{7 ! 5 !} by canceling 7! and then checking divisibility by 5! .$	$\frac{12 !}{7 ! 5 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{7 ! \times 5 !} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$We factor numerator and denominator to cancel common factors$	$\frac{12 \times 11 \times 10 \times 9 \times 8}{5 !} = \frac{2 \times 6 \times 11 \times 2 \times 5 \times 3 \times 3 \times 2 \times 4}{5 \times 4 \times 3 \times 2 \times 1} After cancellation we are left with: 2 \times 3 \times 2 \times 11 which is an integer$	Theory & Tactics Method Card NT1 Click to view full details

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Final Answer

E. I, II, and III are all integers.

Question

Which of the following is an integer? I. \frac{12 !}{6 !} II. \frac{12 !}{8 !} III. \frac{12 !}{7 ! 5 !}

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
$Statement I: We simplify \frac{12 !}{6 !} by canceling the common factorial.$	$\frac{12 !}{6 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 !}{6 !} = 12 \times 11 \times 10 \times 9 \times 8 \times 7$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since the result of Statement I is a product of integers, it is an integer.$	$12 \times 11 \times 10 \times 9 \times 8 \times 7 is an integer$	Theory & Tactics Method Card NT1 Click to view full details
$Statement II: We simplify \frac{12 !}{8 !} by canceling the common factorial.$	$\frac{12 !}{8 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 !}{8 !} = 12 \times 11 \times 10 \times 9$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since the result of Statement II is a product of integers, it is an integer.$	$12 \times 11 \times 10 \times 9 is an integer$	Theory & Tactics Method Card NT1 Click to view full details
$Statement III: We simplify \frac{12 !}{7 ! 5 !} by canceling 7! and then checking divisibility by 5! .$	$\frac{12 !}{7 ! 5 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{7 ! \times 5 !} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$We factor numerator and denominator to cancel common factors$	$\frac{12 \times 11 \times 10 \times 9 \times 8}{5 !} = \frac{2 \times 6 \times 11 \times 2 \times 5 \times 3 \times 3 \times 2 \times 4}{5 \times 4 \times 3 \times 2 \times 1} After cancellation we are left with: 2 \times 3 \times 2 \times 11 which is an integer$	Theory & Tactics Method Card NT1 Click to view full details

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Final Answer

E. I, II, and III are all integers.