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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 27 Jan 2012, 23:32
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Question Stats:

70% (02:51) correct 30% (02:55) wrong based on 264 sessions
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.

[Reveal] Spoiler:
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.
[Reveal] Spoiler: OA

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MGMAT 1 --> 530
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MGMAT 3 ---> 610
GMAT ==> 730


Last edited by Bunuel on 07 Sep 2012, 09:58, edited 2 times in total.
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Re: K&L four digit positive integers [#permalink] New post 28 Jan 2012, 01:53
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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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2 KUDOS received
Director
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Status: Finally Done. Admitted in Kellogg for 2015 intake
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Kudos [?]: 999 [2] , given: 217

Re: K and L are each four-digit positive integers with thousands [#permalink] New post 28 Jan 2012, 15:14
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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, 03:27
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, 03:32
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 21 Jun 2013, 02:47
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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 27 Oct 2014, 07:03
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Re: K and L are each four-digit positive integers with thousands   [#permalink] 27 Oct 2014, 07:03
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