K and L are each four-digit positive integers with thousands

Question

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.

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D

(A) 16
(B) 20
(C) 25
(D) 40
(E) It cannot be determined from the information given.

Bunuel · Accepted Answer

\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.

lagomez · Answer

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

flgators519 · Answer

where did the 400 come from? Thanks

Bunuel · Answer

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.

flgators519 · Answer

thanks guys, perfectly clear

ctrlaltdel · Answer

Thanks Bunuel & lagomez

nickk · Answer

D

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

MHIKER · Answer

Another same at  https://gmatclub.com/forum/function-f-100847.html#p1033076  solved by bunnuel.

nachobioteck · Answer

How are you guys solving these kind of problems... it took me 1.45 just to get a grasp of what the problem wanted and the info given. and then another 2 minutes to solve it... that is way to long. Any insights?

EMPOWERgmatRichC · Answer

HI nachobioteck,

This question is far more "layered" than a typical GMAT Quant question. While you will face some "long" questions on Test Day, you won't see that many. To that end though, "your way" of handling this question is likely a big factor in how long you took to solve it.

Here are some things to keep in mind when approaching any GMAT question:

1) It was written with patterns in mind. The numbers are NOT random, the wording is NOT random - there's at least 1 pattern in it somewhere, so right from the moment you start reading, you need to be looking for that/those pattern(s).

2) You don't have to read a question twice to start taking notes on it.

For example, here's the first half of the first sentence in this prompt (after reading it ONE time, what notes could you take?):

"K and L are each four-digit positive integers with thousands, hundreds, tens, and units digits..."

I would write down..

4-digit numbers
K = _ _ _ _ 
L = _ _ _ _

Now, here's the second half of the first sentence (what notes would you ADD?):

"....as a, b, c, and d, respectively, for the number K, and p, q, r, and s, respectively, for the number L."

I would write ADD...

4-digit numbers
K = a b c d
L = p q r s

Looking at this, it seems pretty straight-forward, but here are the BENEFITS of taking these notes now:
1) I don't have to read the first sentence EVER again.
2) I have a framework for whatever "steps" come next.
3) I can see from the first sentence that this is a THICK question, so I'm on "alert" to pay really careful attention to whatever details come next.

The other explanations in this thread properly present the math involved, so I won't rehash any of that here. The rest of the question is based on spotting prime factorization, knowing your exponent rules and doing a bit of arithmetic.

As you continue to study, remember that every question that you face on the GMAT was BUILT and that GMAT question writers don't have much of an imagination - they have a list of concepts and rules that they have to test you on. While it's a big list, it is also a LIMITED list of possibilities. Look for clues/patterns that remind you of things that you know and you'll be able to speed up even more.

GMAT assassins aren't born, they're made,
Rich

iNumbv · Answer

Why is b-q 4?
Why do you set every other 1? What rule am I missing here?

EMPOWERgmatRichC · Answer

Hi iNumbv,

In the given fraction, both the numerator and denominator have the same "base" numbers (5, 2, 7 and 3) raised to an exponent. Since the fraction = 16, the ONLY base that can factor into that result is the 2, so we have to focus on THOSE exponents attached to the 2s (the b and the q). All of the other variables are inconsequential to the calculation, so making them all "1" allows you to quickly eliminate them from the calculation (since 5^1/5^1 = 1, 3^1/3^1 = 1, etc.).

We're then left with:

2^b/2^q = 2^4

Using exponent rules, this means that b -q = 4

GMAT assassins aren't born, they're made,
Rich

testcracker · Answer

Just brilliant I wish I had such math proficiency as yours

Hanzel101 · Answer

Hi  Bunuel

In this question, I was thrown off "function W" and "function Z"

When I was trying down my notes 
I got f(W) = 5^a*2^b*7^c*3^d ÷ 5^p*2^q*7^r*3^s = f(16) and then I was lost.

How do you decide not to use f(W) instead just equate it to 16?

Please help!

KarishmaB · Answer

K is abcd and L is pqrs.

Given \frac{5^a*2^b*7^c*3^d}{5^p*2^q*7^r*3^s} = 16 = 2^4

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

So all exponents other than (b - q) must be 0 because other factors all become 1. Only (b - q) must be 4.
Hence, a = p, c = r and d = s, but b must be 4 more than q.

So the numbers K and L would look like 3 7 61 and 3 3 61 or 9650 and 9250 etc. When you subtract them, you will always get 400.

(K – L) ÷ 10 = 400/10 = 40

Answer (D)

eeshajain · Answer

same question.

the question stems clearly says f(w)= 5^a*2^b*7^c*3^d ÷ 5^p*2^q*7^r*3^s, w=16.
my understanding of the stem was f(16)= 5^a*2^b*7^c*3^d ÷ 5^p*2^q*7^r*3^s.

Unclear on how the translation of the question stem was "5^a*2^b*7^c*3^d ÷ 5^p*2^q*7^r*3^s=16" instead.

Vishwesh08 · Answer

This is really a good question and definitely not a 605+ level.

axeofflane · Answer

Ngl i solved this by myself and felt like a genius

K and L are each four-digit positive integers with thousands

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