A square grid of squares is constructed so that in the top

Question

A square grid of squares is constructed so that in the top row, each square is darkened, and in the nth row, every nth square is darkened, as shown above. A square grid with this colouring pattern in which no column has more than 10 shaded squares is known in Greenburgh as a â€˜Tara Gridâ€™. An example of such a grid is shown above. Approximately how many squares does the largest â€˜Tara Gridâ€™ have?

(A) 5,000 (B) 10,000 (C) 20,000 (D) 30,000 (E) none of the above

jaypshah · Answer

Nice problem, took me while to figure out even how to begin. I really doubt one would come across such a question on the actual GMAT.

I would go with E which I arrived by trial and error. This the the way I approached it. The aim is to find the smallest LCM of ten numbers as the numbers keep on increasing. 

By this method I got the number of rows as 48. I could have missed a lower LCM but unlikely. Hence squares = approx 50^ 2 = 2500

OA?

kevincan · Answer

Why couldn't this question be on the GMAT? Number properties are heavily tested on thet GMAT. Why did you choose 48 as the maximum number of rows an columns?

primeminister · Answer

I was trying to figure out whether there was a patern visually...and if you look at the right side of the grid it seems like the number of rows when colored squares "overlap", looking from top to bottom, is something like 2^2 then 2^3 then 2^4...etc. It says you need no more than 10 colored squares, so I would take it to 2^10 but subtract one, because across the top you have every square colored. I would think you would need 2^9 rows, ie 512...and that would give 260 thousand plus squares...I would think the answer is E. (please let me know if this is complete crap, I half expect it to be)

All I know for sure is if this were the second question on the quant, I would look around to see if anyone noticed the stench emanating from my full diapers and I would get up and live. 

Any thoughts Kevin?

jaypshah · Answer

The reason I thought that this type of question would be unlikely to come is because it requires more than 3 - 4 steps to arrive at the answer, whereas your normal GMAT questions can be done in 2 - 3 steps.

I got 48 as the LCM of 1,2,3,4,6,8,12,16,24,48 which according to in the smallest, hence corresponds to the maximum rows.

What's the OA on this? Any OE?

kevincan · Answer

You are right in that if there is a number k that is a factor of more than 10 numbers less than or equal to m, then we cannot build a Tara square with m rows. What is the smallest number that has more than 10 factors?

Remember that the best way to count factors of a number is to look at its prime factorization. A number with 11 factors will look like p^10, and a number with 12 factors will look like p^11 or p^3*q^2 where p and q are distinct prime numbers

if p=2, p^10 is more than 1000

But if p=2 and q=3, p^3*q^2=72 (switching q and p would yield a higher number) 

72 has twelve factors and thus a Tara square cannot have 72 rows.
But every integer less than 72 has at most 10 factors, so 71 is the largest possible number of rows of a Tara square, yielding 71^2 squares approx 5000

primeminister · Answer

" Remember that the best way to count factors of a number is to look at its prime factorization. A number with 11 factors will look like p^10, and a number with 12 factors will look like p^11 or p^3*q^2 where p and q are distinct prime numbers"

humbling...care to explain some more?
This may be basic factorization but it escapes me... I understand the part where you look at the problem and say I need the lowest number possible with 10 factors...but from that to 72?...it looks like you didn't just do trial and error. Care to elaborate on the thought process?

Thanks a lot.

A square grid of squares is constructed so that in the top

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