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Senior Manager  Status: Finally Done. Admitted in Kellogg for 2015 intake
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K and L are each four-digit positive integers with thousands  [#permalink]

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46 00:00

Difficulty:   55% (hard)

Question Stats: 69% (02:36) correct 31% (02:47) wrong based on 590 sessions

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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.

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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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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Posts: 59089
Re: K&L four digit positive integers  [#permalink]

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19
19
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.

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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Senior Manager  Status: Finally Done. Admitted in Kellogg for 2015 intake
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Re: K and L are each four-digit positive integers with thousands  [#permalink]

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2
Bunuel thanks - you super star.
_________________
Best Regards,
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MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730
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Re: K&L four digit positive integers  [#permalink]

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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.

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]

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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]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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GMAT 1: 760 Q50 V44 K and L are each four-digit positive integers with thousands  [#permalink]

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2
2
[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]

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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]

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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. Re: K and L are each four-digit positive integers with thousands   [#permalink] 18 Mar 2019, 14:17
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