product of consecutive integers

[drtfyghujd403]

The integer 6 is the product of two consecutive integers (6 = 2 × 3) and the product of three consecutive integers (6 = 1 × 2 × 3). What is the next integer greater than 6 that is both the product of two consecutive integers and the product of three consecutive integers?

A) 153
B) 210
C) 272
D) 336
E) 600

Is there an algebraic way to solve this problem?

[RonPurewal]

there's not going to be an algebraic solution, since algebra doesn't contain tools that deal effectively with concepts like these.

[RonPurewal]

still, if you're going to multiply three consecutive integers together... well, let's just say that the integers aren't going to be very big.
if you put this problem in front of me, i'd just start multiplying together sets of 3 consecutive integers.
1•2•3 = 6, smaller than any of the choices
2•3•4 = 24, smaller than any of the choices
3•4•5 = 60, smaller than any of the choices
4•5•6 = 120, smaller than any of the choices
5•6•7 = 210, hey!
...can i divide 210 into two consecutive integers?
well, i know that 7 has to go into it, and 6•7 and 7•8 are too small.
so... 13•14? nah.
14•15? yep.
done.

that's not so bad.

what does the answer key do?

[RonPurewal]

by the way, don't forget that the content of the exam is specifically designed to punish individuals who want to use ONLY "textbook" methods.

that's the primary reaason for the inclusion of "number properties" problems (as well as other types of problems, mostly DS, that don't submit to algebraic analysis). they want to make sure that the top scores go to flexible thinkers, not to "math jocks".