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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 15 Nov 2009, 11:58
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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.

Source: Manhattan.
[Reveal] Spoiler: OA
D
[Reveal] Spoiler: OA
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Re: Number Properties [#permalink] New post 15 Nov 2009, 12:24
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ctrlaltdel 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.

Source: Manhattan.
[Reveal] Spoiler: OA
D



\frac{5^a*2^b*7^c*3^d}{5^p*2^q*7^r*3^s}=16

5^{a-p}*2^{b-q}*7^{c-r}*3^{d-s}=2^4

a=p, b-q=4, c=r, d=s.

K-L={abcd}-{pqrs}=400

F(Z)=\frac{K-L}{10}=\frac{400}{10}=40

Answer: D.
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Re: Number Properties [#permalink] New post 15 Nov 2009, 16:42
where did the 400 come from? Thanks
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Re: Number Properties [#permalink] New post 15 Nov 2009, 16:54
Expert's post
flgators519 wrote:
where did the 400 come from? Thanks


We got that in two 4 digit numbers abcd and pqrs all numbers except b and q are the same and b-q=4. abcd=1000a+100b+10c+d and pqrs=1000p+100q+10r+s. abcd-pqrs=1000a+100b+10c+d-(1000p+100q+10r+s) as a=p, c=r and d=s, we'll get 100b-100q=(b-q)*100=4*100=400.

Hope it's clear.
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Re: Number Properties [#permalink] New post 15 Nov 2009, 16:57
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Bunuel wrote:
ctrlaltdel 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.

Source: Manhattan.
[Reveal] Spoiler: OA
D





\frac{5^a*2^b*7^c*3^d}{5^p*2^q*7^r*3^s}=16

5^{a-p}*2^{b-q}*7^{c-r}*3^{d-s}=2^4

a=p, b-q=4, c=r, d=s.

K-L={abcd}-{pqrs}=400

F(Z)=\frac{K-L}{10}=\frac{400}{10}=40

Answer: D.


another way to look at it


\frac{5^a*2^b*7^c*3^d}{5^p*2^q*7^r*3^s}=16

Well I know that 2^4 = 16 so I want 2^b/2^q = 2^4

So I set every other number to 1 and b-q should = 4
a = 1; b = 6; c = 1; d = 1
p = 1; q = 2; r = 1; s = 1

1611-1211 = 400/10 = 40
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Re: Number Properties [#permalink] New post 15 Nov 2009, 20:21
thanks guys, perfectly clear
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Re: Number Properties [#permalink] New post 15 Nov 2009, 21:03
Thanks Bunuel & lagomez
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Re: Functions - concepts testing [#permalink] New post 03 Mar 2010, 03:35
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Again, I think my way is kinda slow but here goes:

K = 1000a + 100b + 10c + d
L = 1000p + 100q + 10r + s

W = (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) = 16

Next you do a prime factorisation of 16 and get 2^4
Thus, we know that 2 is the only prime factor of 16, and since the other numbers in W are also prime, we can deduce the following:
a-p = 0
b-q = 4
c-r = 0
d-s = 0

Keeping the statements above in mind, when we evaluate K-L we get:
(100b-100q)/10 = Z
(100(b-q))/10 = Z
(100*4)/10 = Z = 40
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Re: Functions - concepts testing [#permalink] New post 03 Mar 2010, 07:03
IMO D = 40

K = 1000a + 100b + 10c + d
L = 1000p + 100q + 10r + s

also W = 16 = 2^4 now when u equate this value

u will get a=p c=r d=s and b-q = 4 as u will have to equate the powers.

now put that in value of z you will get 10*4 = 40
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Re: Functions - concepts testing [#permalink] New post 23 Jan 2012, 11:52
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 09 Nov 2013, 18:53
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 27 Dec 2013, 15:39
Intimidating, but simple in the end. Kind of long though.
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Re: K and L are each four-digit positive integers with thousands [#permalink] New post 30 Dec 2013, 12:41
ctrlaltdel 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.

Source: Manhattan.
[Reveal] Spoiler: OA
D


The difference of 16 means that b-q = 4

So then, since b-q are on the hundreds digit then the difference is 400

K-L = 400 so 400/10 is 40

Hence, D is the correct answer

Hope it helps
Cheers!
J :)
Re: K and L are each four-digit positive integers with thousands   [#permalink] 30 Dec 2013, 12:41
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