<p>Which of the following could equal (t - 3)(t - 4), if t is a positive even integer?
(A) 50
(B) 42
(C) 32
(D) 22
(E) 12</p>
<p>The answer’s (B) but when i got the explanation, it said to use guess and check.
is there another way besides plugging in numbers?</p>
<p>Use your math “intuition.” (t-3) and (t-4) are 1 apart. 42 = 6*7, t = 10</p>
<p>or:</p>
<p>t cannot be 1, 2, 3, or 4 because all of these yield negative numbers or 0 when you plug them into t-3 or t-4. If both terms are negative, it could allow the product to be positive, but none of the answer choices are small enough besides (E), which is made from (-3)*(-4), but that would imply that t is 0, which is not a positive even integer.</p>
<p>The factors of 50 are 1, 2, 5, 10, 25, and 50, none of which are 1 apart
The factors of 32 are 1, 2, 4, 8, 16, 32, none of which are 1 apart
The factors of 22 are 1, 2, 11, and 22, none of which are 1 apart
The factors of 12 are 1, 2, 3, 4, 6, and 12, none of which are 1 apart</p>
<p>When I say “none of which,” I mean excluding 1, 2, 3, and 4</p>
<p>The easiest way to do this is through INTUITION. You should know in your head that 42 = 6 x 7 and that 12 = 3 x 4. EDIT: read the post below</p>
<p>Exactly…almost. </p>
<p>Since (t-3) and (t-4) are consecutive integers, you need a number that can be written as the product of consecutive integers.</p>
<p>Of course, 12=3*4, but in order to get 3 and 4 from (t-3) and (t-4), you’d need for t to be 7, and 7 is not an even integer. So the answer is 42. Just like in Hitchhiker’s Guide to the Galaxy.</p>
<p>(I have to concede that I completely overlooked (-3)(-4)=12, but as crazybandit points out, that’s ruled out because t cannot equal 0.)</p>