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

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27 Jan 2012, 23:32

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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.

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).

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).

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

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27 Oct 2014, 07:03

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

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29 Nov 2015, 21:22

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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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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