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If the smaller of two consecutive even integers is a multiple of 6, which of the following could NOT be the sum of these two integers?

Explanation	Calculations	Help
$We translate "two consecutive even integers" by letting the smaller be 2 n and the next be 2 n + 2 .$	$2 n, 2 n + 2$	Theory & Tactics Method Card TRANS5-A Click to view full details
$We translate "the smaller is a multiple of 6" by writing 2 n = 6 m, where m is an integer.$	$2 n = 6 m$	Theory & Tactics Method Card TRANS5-E Click to view full details
$We divide both sides of the equation 2 n = 6 m by 2 to find n in terms of m .$	$n = 3 m$
$We translate "the sum of these two integers" by forming their sum.$	$Sum = 2 n + (2 n + 2)$
$We substitute n = 3 m into the sum and simplify.$	$Sum = 2 (3 m) + (2 (3 m) + 2) = 6 m + (6 m + 2) = 12 m + 2$
$We test the answer choices and observe that a sum of the form 12 m + 2 always has remainder 2 when divided by 12. Option B is 30, and dividing 30 by 12 gives a remainder of 6, so 30 cannot be written as 12 m + 2 .$	$30 = 12 \times 2 + 6$

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B

If the smaller of two consecutive even integers is a multiple of 6, which of the following could NOT be the sum of these two integers?

Explanation	Calculations	Help
$We translate "two consecutive even integers" by letting the smaller be 2 n and the next be 2 n + 2 .$	$2 n, 2 n + 2$	Theory & Tactics Method Card TRANS5-A Click to view full details
$We translate "the smaller is a multiple of 6" by writing 2 n = 6 m, where m is an integer.$	$2 n = 6 m$	Theory & Tactics Method Card TRANS5-E Click to view full details
$We divide both sides of the equation 2 n = 6 m by 2 to find n in terms of m .$	$n = 3 m$
$We translate "the sum of these two integers" by forming their sum.$	$Sum = 2 n + (2 n + 2)$
$We substitute n = 3 m into the sum and simplify.$	$Sum = 2 (3 m) + (2 (3 m) + 2) = 6 m + (6 m + 2) = 12 m + 2$
$We test the answer choices and observe that a sum of the form 12 m + 2 always has remainder 2 when divided by 12. Option B is 30, and dividing 30 by 12 gives a remainder of 6, so 30 cannot be written as 12 m + 2 .$	$30 = 12 \times 2 + 6$

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B