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If tuvw = 108, and t, u, v, and w are all positive integers

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by gmattesttaker2 » Mon Jun 23, 2014 7:08 am
Hello,

Can you please tell me how to solve this:

If tuvw = 108, and t, u, v, and w are all positive integers such that t > u > v > w, what is the greatest possible value for t - w?

A) 1
B) 16
C) 17
D) 18
E) 19

OA: 17

Thanks a lot,
Sri
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by Brent@GMATPrepNow » Mon Jun 23, 2014 8:08 am
gmattesttaker2 wrote: If tuvw = 108, and t, u, v, and w are all positive integers such that t > u > v > w, what is the greatest possible value for t - w?

A) 1
B) 16
C) 17
D) 18
E) 19
To maximize the value of t - w, we must MAXIMIZE t and MINIMIZE w

First find the prime factorization of 108.
108 = (2)(2)(3)(3)(3)

We can also add a 1 to this product to get: 108 = (1)(2)(2)(3)(3)(3)
So, the smallest value in our product is 1.
So, w = 1

The next smallest value in our product is 2
So, v = 2

The next smallest value in our product is 3
So, u = 3

Now that we've MINIMIZED the values of w, v and u, we have MAXIMIZED the value of t
If w = 1, v = 2 and u = 3, then t = 18

So, t - w = 18 - 1 = [spoiler]17 = C[/spoiler]

Cheers,
Brent
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by kvcpk » Mon Jun 23, 2014 8:25 am
gmattesttaker2 wrote:Hello,

Can you please tell me how to solve this:

If tuvw = 108, and t, u, v, and w are all positive integers such that t > u > v > w, what is the greatest possible value for t - w?

A) 1
B) 16
C) 17
D) 18
E) 19

OA: 17

Thanks a lot,
Sri
To Maximize the difference, W should be 1.
Add 1 to the answer choices and see which one is a factor of 108.
Looking at the options, A is ruled out because 1 cannot be the difference.
between B,C,D and E only C+1 is a factor of 108.

Pick C
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by GMATinsight » Mon Jun 23, 2014 8:38 pm
The greatest value of t-w will be obtained for least value of w (which is 1 as factor of 108) and greatest value of t which we have to understand by factorising 108 into 4 distinct part such that their product is 108 and smallest factor is 1.

Since the smalles factor is 1 and then next is 2 and the next is 3 therefore the biggest factor which could be the value of t will be 108 / (1 x 2 x 3) = 18

Because, 108 = 1 x 2 x 3 x 18

Thefore t-w = 18 - 1 = 17 Answer
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