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If the smaller of two consecutive even integers is divisible by 3, which of the following could NOT be the sum of these two integers?

Explanation	Calculations	Help
$We translate "two consecutive even integers" into algebraic expressions using a variable.$	$2 k, 2 k + 2$	Theory & Tactics Method Card TRANS5-A Click to view full details
$We translate "the smaller is divisible by 3" into a divisibility equation.$	$2 k = 3 n$	Theory & Tactics Method Card TRANS5-E Click to view full details
$We note that the left side is divisible by 2, so n must be even; we express this with a new variable.$	$n = 2 m$	Theory & Tactics Method Card TRANS5-A Click to view full details
$We express the sum of the two integers.$	$Sum = 2 k + (2 k + 2)$
$We substitute the previous expressions into the sum and simplify step by step.$	$Sum = 2 k + (2 k + 2) = 4 k + 2 = 4 (\frac{3 n}{2}) + 2 = 6 n + 2 = 6 (2 m) + 2 = 12 m + 2$
$We observe that the sum must leave a remainder of 2 when divided by 12, so testing the options shows which fails this pattern.$	$76 = 12 \times 6 + 4$

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E

If the smaller of two consecutive even integers is divisible by 3, which of the following could NOT be the sum of these two integers?

Explanation	Calculations	Help
$We translate "two consecutive even integers" into algebraic expressions using a variable.$	$2 k, 2 k + 2$	Theory & Tactics Method Card TRANS5-A Click to view full details
$We translate "the smaller is divisible by 3" into a divisibility equation.$	$2 k = 3 n$	Theory & Tactics Method Card TRANS5-E Click to view full details
$We note that the left side is divisible by 2, so n must be even; we express this with a new variable.$	$n = 2 m$	Theory & Tactics Method Card TRANS5-A Click to view full details
$We express the sum of the two integers.$	$Sum = 2 k + (2 k + 2)$
$We substitute the previous expressions into the sum and simplify step by step.$	$Sum = 2 k + (2 k + 2) = 4 k + 2 = 4 (\frac{3 n}{2}) + 2 = 6 n + 2 = 6 (2 m) + 2 = 12 m + 2$
$We observe that the sum must leave a remainder of 2 when divided by 12, so testing the options shows which fails this pattern.$	$76 = 12 \times 6 + 4$

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E