How Many Sets of Four Consecutive Integers Have a Product Under 100,000?

[rocomath]

How many sets of four consecutive integers are there such that the product of the four integers is less than 100,000?

$$Set_1=1,2,3,4$$
$$Set_2=5,6,7,8$$
$$Set_3=9,10,11,12$$

$$Set_n=a\cdot b\cdot c\cdot d<100,000$$

Okay, I know I could continue with my Sets, but there has got to be a more logical approach. Help start me off please! I've already plugged and chugged it and found the amount of sets, but I want a better approach.

 

[HallsofIvy]

Why did you skip over 2,3, 4, 5? Or 3, 4, 5, 6? Do you see that if, for some a, b, c, d, their product is less than 100000, so is the product of any 4 consecutive integers less than that?

How large must a, b, c, d be so that their product IS 100000? Since a, b, c, d are consecutive, they are relatively close to each other so their product must be close to b⁴. Solve b⁴= 100000 and look for a, b, c, d close to that. Do you see that the number sets of 4 consectutive integers whose product is less than 100000 is simply the "a-1" of the first a, b, c, d whose product is equal to or greater than 100000?

(The answer is surprisingly low!)

 

[rocomath]

Ah crap, I completely forgot about 2,3,4,5 etc :( Now my answer is completely wrong ...

Ok let me go back to this problem.

 

[HallsofIvy]

You knew your answer was not right to begin with- that's why you asked the question!

Have you payed any attention to the rest of my response? It took me about 2 minutes to solve that problem (counting the time I spent sharpening my pencil).