BTGmoderatorDC wrote: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

OA B

Source: Manhattan Prep

This question can better be solved through hit and trial.

We see that 210 can be written as 2*3*5*7; though looking at this, we do not see that there is a possibility of finding 210 as the product of two consecutive integers and the product of three consecutive integers, upon a close look, we can turn 2*3*5*7 into them.

2*3*5*7 can be written as 5*6*7 -- a product of three consecutive integers and
2*3*5*7 can be written as 14*15 -- a product of two consecutive integers.

Thus, 210 is the correct answer.

The correct answer: B

Hope this helps!

-Jay
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BTGmoderatorDC wrote: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

OA B

Source: Manhattan Prep

The product of 3 consecutive integers must be divisible by 6.

We see that 210, 336, and 600 are all divisible by 6 (so we can eliminate 153 and 272). Let's test 210.

Factoring 210, we get: 210 = 10 x 21 = 2 x 5 x 3 x 7

We see that 210 = (2 x 7) x (3 x 5) = 14 x 15 and 210 = 5 x (2 x 3) x 7 = 5 x 6 x 7. Therefore, 210 is the next greater integer that is both the product of two consecutive integers and the product of three consecutive integers.

Answer: B