function f

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function f

by P_mashru » Fri Jun 20, 2008 3:41 am
The function f is defined for each positive three-digit integer n by f(n) = 2(x).3(y).5(z), where x, y & z are the hundreds, tens, and units digits of n, respectively. If m an v are three - digit positive integers such that
f(m)=9f(v), then what could be the value of (m-v)=?

a) 18
b) 19
c) 15
d) 21
e) 80

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by g_beatthegmat » Fri Jun 20, 2008 4:15 am
IMO, the answer is (E) 80.

Is that the OA as well? If yes then I can elaborate on my explanation :)

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by P_mashru » Fri Jun 20, 2008 4:48 am
OA is E only,

Pl elaborate
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Re: function f

by Ian Stewart » Fri Jun 20, 2008 5:01 am
P_mashru wrote:The function f is defined for each positive three-digit integer n by f(n) = 2(x).3(y).5(z), where x, y & z are the hundreds, tens, and units digits of n, respectively.
There is something wrong with the wording of this question, for a start. I wonder whether you mean 2^x * 3^y * 5^z? It looks suspiciously like a prime factorization, which is why I'd expect x, y and z to be exponents here. I'll do both questions below- first I'll assume it's a product, and after, I'll assume x, y and z are exponents. Either way, there is something wrong with the answer choices.

I'll first assume that the question is supposed to be as you've written it- that we are multiplying x, y and z.

Then

f(n) = 30xyz

Let's let m = abc, v = xyz, where a, b, c, x, y and z are digits.

We know
f(m) = 9f(v)
30abc = 9*30xyz
abc = 9xyz

That is, the product of m's digits is 9 times the product of v's digits. That could happen in quite a few different ways; the answer choices may tell us what we're looking for. We want m-v to be a two digit number. So, why not try making the second digit of m = 9 times the second digit of v, and let the other digits remain equal? That is, let the middle digit of v be 1, and the middle digit of m be 9. We'd then have numbers like:

m = 191
v = 111

or

m = 492
v = 412

and in each case, m-v = 80.

So m-v could be 80. Could it be any of the other answer choices? Yes, in fact; it could be all of them. If we allow one digit to be 0, something which is not excluded in the question, we can get any two-digit value for m-v that we like. e.g.

m = 220
v = 202
m-v = 18

Notice the product of m's digits is still 9 times the product of v's digits, since 9*0 = 0, so in the above case, f(m) = 9*f(v). So, as the question is worded, all five answers are correct. Either x, y and z are intended to be exponents, or the question is incorrectly worded. I'm curious- where is this question from?

I'd note, finally, that if x, y and z are intended to be exponents, (which I suspect is what's really supposed to be happening here) then we would know (again letting m = abc, v = xyz):

f(m) = 9*f(v)
2^a * 3^b * 5^c = 3^2 * 2^x * 3^y * 5^z
2^a * 3^b * 5^c = 2^x * 3^(y+2) * 5^z

These are prime factorizations of integers, so a=x, b=y+2, c=z. Thus m and v only differ in their second digit; the second digit of m is two larger than the second digit of v. The difference must be 20, which isn't in the five answer choices you've listed above.

P_mashru wrote: If m an v are three - digit positive integers such that
f(m)=9f(v), then what could be the value of (m-v)=?

a) 18
b) 19
c) 15
d) 21
e) 80

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by P_mashru » Fri Jun 20, 2008 7:38 pm
Ian,

Hats off to your ability to explain things.. I am taking GMAT coaching from Endevour in ahmedabad, India & this question appeared in one of our sectional test.

I will take up your solution to my quant coach in doubt clearing session tomorrow and update about solution from their side,

thnax
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by beeparoo » Sat Jun 21, 2008 5:48 pm
P_mashru wrote:Ian,

Hats off to your ability to explain things.. I am taking GMAT coaching from Endevour in ahmedabad, India & this question appeared in one of our sectional test.

I will take up your solution to my quant coach in doubt clearing session tomorrow and update about solution from their side,

thnax
:shock: OMG - are you kidding me? My brain exploded trying to solve this nutty problem (before venturing to scroll down to check the answers).

There are SO MANY people and posted questions that do not type up the question properly. While it is often frustrating, I admit I've learned to separate genuine gmat questions from non-genuine ones.

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Typo in the answer choice

by venkatsr » Sun Sep 13, 2009 4:25 pm
A. 8
B. 9
C. 18
D. 20
E. 80

Ian was able crack it easily anyway.