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K and L are each four-digit positive integers with thousands

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K and L are each four-digit positive integers with thousands [#permalink] New post 15 Nov 2009, 11:58
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K and L are each four-digit positive integers with thousands, hundreds, tens, and units digits defined as a, b, c, and d, respectively, for the number K, and p, q, r, and s, respectively, for the number L. For numbers K and L, the function W is defined as 5^a*2^b*7^c*3^d ÷ 5^p*2^q*7^r*3^s. The function Z is defined as (K – L) ÷ 10. If W = 16, what is the value of Z?

(A) 16
(B) 20
(C) 25
(D) 40
(E) It cannot be determined from the information given.

Source: Manhattan.
[Reveal] Spoiler: OA
D
[Reveal] Spoiler: OA
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Re: Number Properties [#permalink] New post 15 Nov 2009, 12:24
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ctrlaltdel wrote:
K and L are each four-digit positive integers with thousands, hundreds, tens, and units digits defined as a, b, c, and d, respectively, for the number K, and p, q, r, and s, respectively, for the number L. For numbers K and L, the function W is defined as 5^a*2^b*7^c*3^d ÷ 5^p*2^q*7^r*3^s. The function Z is defined as (K – L) ÷ 10. If W = 16, what is the value of Z?

(A) 16
(B) 20
(C) 25
(D) 40
(E) It cannot be determined from the information given.

Source: Manhattan.
[Reveal] Spoiler: OA
D



\(\frac{5^a*2^b*7^c*3^d}{5^p*2^q*7^r*3^s}=16\)

\(5^{a-p}*2^{b-q}*7^{c-r}*3^{d-s}=2^4\)

\(a=p\), \(b-q=4\), \(c=r\), \(d=s\).

\(K-L={abcd}-{pqrs}=400\)

\(F(Z)=\frac{K-L}{10}=\frac{400}{10}=40\)

Answer: D.
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Re: Number Properties [#permalink] New post 15 Nov 2009, 16:42
where did the 400 come from? Thanks
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Re: Number Properties [#permalink] New post 15 Nov 2009, 16:54
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flgators519 wrote:
where did the 400 come from? Thanks


We got that in two 4 digit numbers abcd and pqrs all numbers except b and q are the same and b-q=4. abcd=1000a+100b+10c+d and pqrs=1000p+100q+10r+s. abcd-pqrs=1000a+100b+10c+d-(1000p+100q+10r+s) as a=p, c=r and d=s, we'll get 100b-100q=(b-q)*100=4*100=400.

Hope it's clear.
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Re: Number Properties [#permalink] New post 15 Nov 2009, 16:57
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Bunuel wrote:
ctrlaltdel wrote:
K and L are each four-digit positive integers with thousands, hundreds, tens, and units digits defined as a, b, c, and d, respectively, for the number K, and p, q, r, and s, respectively, for the number L. For numbers K and L, the function W is defined as 5^a*2^b*7^c*3^d ÷ 5^p*2^q*7^r*3^s. The function Z is defined as (K – L) ÷ 10. If W = 16, what is the value of Z?

(A) 16
(B) 20
(C) 25
(D) 40
(E) It cannot be determined from the information given.

Source: Manhattan.
[Reveal] Spoiler: OA
D





\(\frac{5^a*2^b*7^c*3^d}{5^p*2^q*7^r*3^s}=16\)

\(5^{a-p}*2^{b-q}*7^{c-r}*3^{d-s}=2^4\)

\(a=p\), \(b-q=4\), \(c=r\), \(d=s\).

\(K-L={abcd}-{pqrs}=400\)

\(F(Z)=\frac{K-L}{10}=\frac{400}{10}=40\)

Answer: D.


another way to look at it


\(\frac{5^a*2^b*7^c*3^d}{5^p*2^q*7^r*3^s}=16\)

Well I know that 2^4 = 16 so I want 2^b/2^q = 2^4

So I set every other number to 1 and b-q should = 4
a = 1; b = 6; c = 1; d = 1
p = 1; q = 2; r = 1; s = 1

1611-1211 = 400/10 = 40
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Re: Number Properties [#permalink] New post 15 Nov 2009, 20:21
thanks guys, perfectly clear
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Re: Number Properties [#permalink] New post 15 Nov 2009, 21:03
Thanks Bunuel & lagomez
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Re: Functions - concepts testing [#permalink] New post 03 Mar 2010, 03:35
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Again, I think my way is kinda slow but here goes:

K = 1000a + 100b + 10c + d
L = 1000p + 100q + 10r + s

W = (5^a 2^b 7^c 3^d)/(5^p 2^q 7^r 3^s) = 16
W = 5^(a-p) 2^(b-q) 7^(c-r) 3(d-s) = 16

Next you do a prime factorisation of 16 and get 2^4
Thus, we know that 2 is the only prime factor of 16, and since the other numbers in W are also prime, we can deduce the following:
a-p = 0
b-q = 4
c-r = 0
d-s = 0

Keeping the statements above in mind, when we evaluate K-L we get:
(100b-100q)/10 = Z
(100(b-q))/10 = Z
(100*4)/10 = Z = 40
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Re: Functions - concepts testing [#permalink] New post 03 Mar 2010, 07:03
IMO D = 40

K = 1000a + 100b + 10c + d
L = 1000p + 100q + 10r + s

also W = 16 = 2^4 now when u equate this value

u will get a=p c=r d=s and b-q = 4 as u will have to equate the powers.

now put that in value of z you will get 10*4 = 40
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Re: Functions - concepts testing [#permalink] New post 23 Jan 2012, 11:52
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 09 Nov 2013, 18:53
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 27 Dec 2013, 15:39
Intimidating, but simple in the end. Kind of long though.
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 30 Dec 2013, 12:41
ctrlaltdel wrote:
K and L are each four-digit positive integers with thousands, hundreds, tens, and units digits defined as a, b, c, and d, respectively, for the number K, and p, q, r, and s, respectively, for the number L. For numbers K and L, the function W is defined as 5^a*2^b*7^c*3^d ÷ 5^p*2^q*7^r*3^s. The function Z is defined as (K – L) ÷ 10. If W = 16, what is the value of Z?

(A) 16
(B) 20
(C) 25
(D) 40
(E) It cannot be determined from the information given.

Source: Manhattan.
[Reveal] Spoiler: OA
D


The difference of 16 means that b-q = 4

So then, since b-q are on the hundreds digit then the difference is 400

K-L = 400 so 400/10 is 40

Hence, D is the correct answer

Hope it helps
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 03 Jan 2015, 10:34
How are you guys solving these kind of problems... it took me 1.45 just to get a grasp of what the problem wanted and the info given. and then another 2 minutes to solve it... that is way to long. Any insights?
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 03 Jan 2015, 11:26
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HI nachobioteck,

This question is far more "layered" than a typical GMAT Quant question. While you will face some "long" questions on Test Day, you won't see that many. To that end though, "your way" of handling this question is likely a big factor in how long you took to solve it.

Here are some things to keep in mind when approaching any GMAT question:

1) It was written with patterns in mind. The numbers are NOT random, the wording is NOT random - there's at least 1 pattern in it somewhere, so right from the moment you start reading, you need to be looking for that/those pattern(s).

2) You don't have to read a question twice to start taking notes on it.

For example, here's the first half of the first sentence in this prompt (after reading it ONE time, what notes could you take?):

"K and L are each four-digit positive integers with thousands, hundreds, tens, and units digits..."

I would write down..

4-digit numbers
K = _ _ _ _
L = _ _ _ _

Now, here's the second half of the first sentence (what notes would you ADD?):

"....as a, b, c, and d, respectively, for the number K, and p, q, r, and s, respectively, for the number L."

I would write ADD...

4-digit numbers
K = a b c d
L = p q r s

Looking at this, it seems pretty straight-forward, but here are the BENEFITS of taking these notes now:
1) I don't have to read the first sentence EVER again.
2) I have a framework for whatever "steps" come next.
3) I can see from the first sentence that this is a THICK question, so I'm on "alert" to pay really careful attention to whatever details come next.

The other explanations in this thread properly present the math involved, so I won't rehash any of that here. The rest of the question is based on spotting prime factorization, knowing your exponent rules and doing a bit of arithmetic.

As you continue to study, remember that every question that you face on the GMAT was BUILT and that GMAT question writers don't have much of an imagination - they have a list of concepts and rules that they have to test you on. While it's a big list, it is also a LIMITED list of possibilities. Look for clues/patterns that remind you of things that you know and you'll be able to speed up even more.

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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 04 Jan 2015, 18:07
Kudos for you! Thank you very much that was very insightful. I definitely have to improve my patter recognition and work on my strategies, the math I have it down just need to improve a little my speed.... Again thanks a lot!
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K and L are each four-digit positive integers with thousands [#permalink] New post 12 Jan 2015, 07:02
lagomez wrote:
Bunuel wrote:
ctrlaltdel wrote:
K and L are each four-digit positive integers with thousands, hundreds, tens, and units digits defined as a, b, c, and d, respectively, for the number K, and p, q, r, and s, respectively, for the number L. For numbers K and L, the function W is defined as 5^a*2^b*7^c*3^d ÷ 5^p*2^q*7^r*3^s. The function Z is defined as (K – L) ÷ 10. If W = 16, what is the value of Z?

(A) 16
(B) 20
(C) 25
(D) 40
(E) It cannot be determined from the information given.

Source: Manhattan.
[Reveal] Spoiler: OA
D





\(\frac{5^a*2^b*7^c*3^d}{5^p*2^q*7^r*3^s}=16\)

\(5^{a-p}*2^{b-q}*7^{c-r}*3^{d-s}=2^4\)

\(a=p\), \(b-q=4\), \(c=r\), \(d=s\).

\(K-L={abcd}-{pqrs}=400\)

\(F(Z)=\frac{K-L}{10}=\frac{400}{10}=40\)

Answer: D.


another way to look at it


\(\frac{5^a*2^b*7^c*3^d}{5^p*2^q*7^r*3^s}=16\)

Well I know that 2^4 = 16 so I want 2^b/2^q = 2^4

So I set every other number to 1 and b-q should = 4
a = 1; b = 6; c = 1; d = 1
p = 1; q = 2; r = 1; s = 1

1611-1211 = 400/10 = 40



Why is b-q 4?
Why do you set every other 1? What rule am I missing here?
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 12 Jan 2015, 12:24
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Hi iNumbv,

In the given fraction, both the numerator and denominator have the same "base" numbers (5, 2, 7 and 3) raised to an exponent. Since the fraction = 16, the ONLY base that can factor into that result is the 2, so we have to focus on THOSE exponents attached to the 2s (the b and the q). All of the other variables are inconsequential to the calculation, so making them all "1" allows you to quickly eliminate them from the calculation (since 5^1/5^1 = 1, 3^1/3^1 = 1, etc.).

We're then left with:

2^b/2^q = 2^4

Using exponent rules, this means that b -q = 4

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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 13 Jan 2015, 22:42
K = abcd

L = pqrs

\(16 = \frac{5^a * 2^b * 7^c * 3^d}{5^p * 2^q * 7^r * 3^s} = 2^4\)

\(5^a * 2^b * 7^c * 3^d = 5^p * 2^{(q+4)} * 7^r * 3^s\)

Equating LHS with RHS

a = p; b = q+4; c = r; d = s

Let a = p = 7; b = 6 & q = 2; c = r = 5; d = s = 4

K = 7654 & L = 7254

\(\frac{K-L}{10} = \frac{7654 - 7254}{10} = \frac{400}{10} = 40\)

Answer = D
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Re: K and L are each four-digit positive integers with thousands   [#permalink] 13 Jan 2015, 22:42
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