A function V(a, b) is defined for positive integers a, b

Expert replies

Previous Topic Next Topic

M7MBA: Moderator; Posts: 2058; Joined: Sun Oct 29, 2017 4:24 amFollowed by: 5 members; Thanked: 1 times

by M7MBA » Sat May 12, 2018 6:09 am

Timer

00:00

Answers

Stats

Difficulty—

A function V(a, b) is defined for positive integers a, b and satisfies

V(a, a) = a,
V(a, b) = V(b, a),
V(a, a+b) = (1 + a/b) V(a, b).

The value represented by V(66, 14) is?

(A) 364
(B) 231
(C) 455
(D) 472
(E) None of the foregoing

The OA is the option E.

Can anyone help me here? I need an explanation here, please. I couldn't solve it.

Like this post

Source: Beat The GMAT — Problem Solving | Sat May 12, 2018 6:09 am

Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

answer

by Vincen » Fri May 18, 2018 5:30 am

Hello M7MBA.

This is a little long question. Here is how I'd solve it.

V(a, a) = a,
V(a, b) = V(b, a),
V(a, a+b) = (1 + a/b) V(a, b).

Using the second property we have $$V(66,14)=V\left(14,66\right)$$ Now, using the third property many times we get: $$V\left(14,66\right)=V\left(14,\ 14+52\right)=\left(1+\frac{14}{52}\right)V\left(14,52\right)=\frac{33}{26}V\left(14,52\right).$$ $$V\left(14,52\right)=V\left(14,\ 14+38\right)=\frac{26}{19}V\left(14,38\right).$$ $$V\left(14,38\right)=V\left(14,\ 14+24\right)=\frac{19}{12}V\left(14,24\right)$$ $$V\left(14,24\right)=V\left(14,14+10\right)=\frac{12}{5}V\left(14,10\right).$$ $$V\left(14,10\right)=V\left(10,14\right)=V\left(10,\ 10+4\right)=\frac{7}{2}V\left(10,4\right).$$ $$V\left(10,4\right)=V\left(4,10\right)=V\left(4,4+6\right)=\frac{5}{3}V\left(4,6\right).$$ $$V\left(4,6\right)=V\left(4,4+2\right)=3V\left(4,2\right).$$ $$V\left(4,2\right)=V\left(2,4\right)=V\left(2,2+2\right)=2V\left(2,2\right)=2\cdot2=4.$$ Now, we have to replace all the results as follows: $$V\left(14,66\right)=\frac{33}{26}\cdot\frac{26}{19}\cdot\frac{19}{12}\cdot\frac{12}{5}\cdot\frac{7}{2}\cdot\frac{5}{3}\cdot3\cdot4=\frac{33\cdot7\cdot4}{2}=33\cdot7\cdot2=462.$$ Since the answer is none of A, B, C or D, then the answer is [spoiler]E=none of the foregoing[/spoiler].

I hope it is clear enough.

Last edited by Vincen on Tue May 22, 2018 12:23 am, edited 1 time in total.

Like this post

Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

by Jeff@TargetTestPrep » Mon May 21, 2018 3:50 pm

M7MBA wrote:A function V(a, b) is defined for positive integers a, b and satisfies

V(a, a) = a,
V(a, b) = V(b, a),
V(a, a+b) = (1 + a/b) V(a, b).

The value represented by V(66, 14) is?

(A) 364
(B) 231
(C) 455
(D) 472
(E) None of the foregoing

First, we need to mention that with the three formulas given, we always want to express V(b, a) as V(a, b) whenever b > a. Thus we can simplify the expression continuously until it reaches a point where the two numbers for the function are the same.

Next, notice that in the last formula, we have: V(a, a+b) = (1 + a/b) V(a, b) = (a+b)/b V(a, b). If we let c = a + b (we see that c > a), then V(a, c) = c/(c - a) V(a, c - a). So we'll be using this formula instead of V(a, a+b) = (1 + a/b) V(a, b) in our calculations that follows. Now let's begin.

Using the second property, we have: V(66, 14) = V(14, 66). So

V(14, 66) = 66/52 V(14, 52) (note: recall that V(a, c) = c/(c - a) V(a, c - a) when c > a)

V(14, 52) = 52/38 V(14, 38)

V(14, 38) = 38/24 V(14, 24)

V(14, 24) = 24/10 V(14, 10) = 24/10 V(10, 14)

V(10, 14) = 14/4 V(10, 4) = 14/4 V(4, 10)

V(4, 10) = 10/6 V(4, 6)

V(4, 6) = 6/2 V(4, 2) = 6/2 V(2, 4)

V(2, 4) = 4/2 V(2, 2)

V(2, 2) = 2

So we have:

V(14, 66) = 66/52 * 52/38 * 38/24 * 24/10 * 14/4 * 10/6 * 6/2 * 4/2 * 2 = 66 * 14 * 1/2 = 462

Answer: E

Jeffrey Miller
Head of GMAT Instruction
[email protected]

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

559 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review