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GMAT Quant — Number Properties

Which of the following expressions is an integer? I. \frac{10 !}{4 ! 6 !} II. \frac{10 !}{3 ! 7 !} III. \frac{10 !}{5 ! 8 !}

Explanation	Calculations	Help
$Statement I: We simplify \frac{10 !}{4 ! 6 !} by canceling the common 6! factor.$	$\frac{10 !}{4 ! 6 !} = \frac{10 \times 9 \times 8 \times 7 \times 6 !}{6 ! \times 4 !} = \frac{10 \times 9 \times 8 \times 7}{4 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since 4! = 24 divides evenly into 10 \times 9 \times 8 \times 7, the expression is an integer.$	$10 \times 9 \times 8 \times 7 \div 4! yields an integer$
$Statement II: We simplify \frac{10 !}{3 ! 7 !} by canceling the common 7! factor.$	$\frac{10 !}{3 ! 7 !} = \frac{10 \times 9 \times 8 \times 7 !}{7 ! \times 3 !} = \frac{10 \times 9 \times 8}{3 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since 3! = 6 divides evenly into 10 \times 9 \times 8, the expression is an integer.$	$10 \times 9 \times 8 \div 3! yields an integer$
$Statement III: We simplify \frac{10 !}{5 ! 8 !} by canceling the common 8! factor.$	$\frac{10 !}{5 ! 8 !} = \frac{10 \times 9 \times 8 !}{8 ! \times 5 !} = \frac{10 \times 9}{5 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$We factor and cancel common factors to see if the result is an integer.$	$\frac{10 \times 9}{5 \times 4 \times 3 \times 2 \times 1} = \frac{2 \times 3}{4 \times 2 \times 1} = \frac{6}{8} = \frac{3}{4}$	Theory & Tactics Method Card FDPR2-A Click to view full details
$Since the result is \frac{3}{4}, which is not a whole number, Statement III is not an integer.$	$\frac{3}{4} is not an integer$

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D

GMAT Quant — Number Properties

Which of the following expressions is an integer? I. \frac{10 !}{4 ! 6 !} II. \frac{10 !}{3 ! 7 !} III. \frac{10 !}{5 ! 8 !}

Explanation	Calculations	Help
$Statement I: We simplify \frac{10 !}{4 ! 6 !} by canceling the common 6! factor.$	$\frac{10 !}{4 ! 6 !} = \frac{10 \times 9 \times 8 \times 7 \times 6 !}{6 ! \times 4 !} = \frac{10 \times 9 \times 8 \times 7}{4 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since 4! = 24 divides evenly into 10 \times 9 \times 8 \times 7, the expression is an integer.$	$10 \times 9 \times 8 \times 7 \div 4! yields an integer$
$Statement II: We simplify \frac{10 !}{3 ! 7 !} by canceling the common 7! factor.$	$\frac{10 !}{3 ! 7 !} = \frac{10 \times 9 \times 8 \times 7 !}{7 ! \times 3 !} = \frac{10 \times 9 \times 8}{3 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$Since 3! = 6 divides evenly into 10 \times 9 \times 8, the expression is an integer.$	$10 \times 9 \times 8 \div 3! yields an integer$
$Statement III: We simplify \frac{10 !}{5 ! 8 !} by canceling the common 8! factor.$	$\frac{10 !}{5 ! 8 !} = \frac{10 \times 9 \times 8 !}{8 ! \times 5 !} = \frac{10 \times 9}{5 !}$	Theory & Tactics Method Card CALC2-C Click to view full details
$We factor and cancel common factors to see if the result is an integer.$	$\frac{10 \times 9}{5 \times 4 \times 3 \times 2 \times 1} = \frac{2 \times 3}{4 \times 2 \times 1} = \frac{6}{8} = \frac{3}{4}$	Theory & Tactics Method Card FDPR2-A Click to view full details
$Since the result is \frac{3}{4}, which is not a whole number, Statement III is not an integer.$	$\frac{3}{4} is not an integer$

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D