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K and L are each four-digit positive integers with thousands
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Updated on: 07 Sep 2012, 10:58
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40
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Difficulty:
55% (hard)
Question Stats:
69% (02:38) correct 31% (02:45) wrong based on 544 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.
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).
K and L are each four-digit positive integers with thousands
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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
Re: K and L are each four-digit positive integers with thousands
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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.
gmatclubot
Re: K and L are each four-digit positive integers with thousands
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18 Mar 2019, 14:17
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69% (02:38) correct 
