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

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enigma123
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K and L are each four-digit positive integers with thousands [#permalink] New post   Updated on: 07 Sep 2012, 10:58
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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.

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As the OA is not given this is how I am trying to solve this question, but I am stuck now. Can someone help please?

Digits of K are a,b,c and d i.e. K is 1000 a + 100 b + 10 c + d ----------------------------------------(1)

Digits of L are p, q, r and s i.e. L is 1000 p + 100 q + 10 r + s ----------------------------------------(2)

f(w) = \(5^a 2^b 7^c 3^d\) / \(5^p 2^q 7^r 3^s\)-----------------------------------------(3)

Also, f(16) = \(5^a 2^b 7^c 3^d\) / \(5^p 2^q 7^r 3^s\)-----------------------------------(4)

Therefore, \(5^a 2^b 7^c 3^d\) / \(5^p 2^q 7^r 3^s\) = 16 i.e. \(2^4\)----------------(5)

f(z) = (1000 a + 100 b + 10 c + d) - 1000 p + 100 q + 10 r + s / 10 ----------------------------------(6)

Now, I am stuck and no OA doesn't help either. Therefore, your help will be much appreciated guys.
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D

Originally posted by enigma123 on 28 Jan 2012, 00:32.
Last edited by Bunuel on 07 Sep 2012, 10:58, edited 2 times in total.
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Re: K&L four digit positive integers [#permalink] New post   28 Jan 2012, 02:53
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enigma123 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.


Given: \(w=\frac{5^a*2^b*7^c*3^d}{5^p*2^q*7^r*3^s}=16\) --> \(w=5^{a-p}*2^{b-q}*7^{c-r}*3^{d-s}=2^4\) --> the powers of 3, 5, and 7 must be zero and the power of 2 must be 4: \(a=p\), \(b-q=4\), \(c=r\) and \(d=s\)

Now, as thousands, tens, and units digits in K and L are equal and the difference between hundreds' digits is 4, then K-L=400 (for example K=1923 and L=1523 --> K-L=1923-1523=400).

Z=(K-L)/10=400/10=40.

Answer: D.

Also discussed here: functions-concepts-testing-91004.html
Similar questions:
the-function-f-is-defined-for-each-positive-three-digit-100847.html
for-any-four-digit-number-abcd-abcd-3-a-5-b-7-c-11-d-126522.html

Hope it helps.
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K and L are each four-digit positive integers with thousands [#permalink] New post   01 Dec 2015, 01:03
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[quote="enigma123"]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.

Given:
K = abcd = 1000a + 100b + 10c + d
L = pqrs = 1000p + 100q + 10r + s
W = \(5^a 2^b 7^c 3^d\) ÷ \(5^p 2^q 7^r 3^s\) = \(5^{a-p} 2^{b-q} 3^{c-r} 5^{d-s}\) = 16 = \(2^4\)

W can 16 only when W carries the powers of 2 only.
Hence b - q = 4 (i)
And the rest of the powers will be 0.
a= p, c = r, d = s (ii)

Required: Z = (K – L) ÷ 10 =?
Z = (abcd - pqrs)÷10 = (1000a + 100b + 10c + d) - (1000p + 100q + 10r + s) ÷ 10
Z = 1000 (a - p) + 100(b - q) + 10 (c - r) + 10 (d - s) ÷ 10
From equations (i) and (ii)
Z = 100(b-q) ÷ 10 = 100*4 ÷ 10= 40
Option D
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post   28 Jan 2012, 16:14
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Bunuel thanks - you super star.
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Re: K&L four digit positive integers [#permalink] New post   14 Sep 2012, 04:27
Bunuel wrote:
enigma123 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.


Given: \(w=\frac{5^a*2^b*7^c*3^d}{5^p*2^q*7^r*3^s}=16\) --> \(w=5^{a-p}*2^{b-q}*7^{c-r}*3^{d-s}=2^4\) --> the powers of 3, 5, and 7 must be zero and the power of 2 must be 4: \(a=p\), \(b-q=4\), \(c=r\) and \(d=s\)

Now, as thousands, tens, and units digits in K and L are equal and the difference between hundreds' digits is 4, then K-L=400 (for example K=1923 and L=1523 --> K-L=1923-1523=400).

Z=(K-L)/10=400/10=40.

Answer: D.

Also discussed here: functions-concepts-testing-91004.html
Similar question: the-function-f-is-defined-for-each-positive-three-digit-100847.html

Hope it helps.



Hi Bunuel ,

Can you move this question to the quant section
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Re: K&L four digit positive integers [#permalink] New post   14 Sep 2012, 04:32
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fameatop wrote:
Hi Bunuel ,

Can you move this question to the quant section


Done: the question is moved to PS forum. Thank you.
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post   21 Jun 2013, 03:47
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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: K and L are each four-digit positive integers with thousands [#permalink] New post   28 Oct 2016, 17:18
This question is tricky in the sense that it is a simple recognition of 2^4=16 that is buried under quite a bit of language.

W can be rewritten to the following: 16=5^(a-p) x 2^(b-q) x 7^(c-r) x 3^(d-s)

K and L are four digit numbers, so we need to have at least 1 in a and p's position. Thus, we will let a=1, p=1

b-q needs to equal 4, so we can assign any single digits that would give us that result. b=9, q=5

The rest of the numbers can be either 0's or 1's. Doesn't matter as we will be subtracting them off when we plug values into the Z equation.

K = 1911
L = 1511

Z = (K-L)/40 = 400/10 = 40
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post   18 Mar 2019, 14:17
Agreed that this question's biggest challenge is the wordiness...

The way I approached it:

1) f(W) shows us the division of the prime factorization of the 2 numbers, K and L. That f(W) = 16 means there's 2^4 more in the numerator K, i.e. that b-4 = q. This also means that all the other variables are equal.

2) f(Z) = (F - L)/10 ... since all the variables are equal except b and q (the hundreds place), it will be some number where the difference between them is 400 (since b is greater than q by 4). 400/10 = 40, D.
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post   18 Aug 2023, 08:05
Hello from the GMAT Club BumpBot!

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Re: K and L are each four-digit positive integers with thousands [#permalink]
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