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

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

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

by M7MBA » Sat May 12, 2018 6:09 am

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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.

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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 pm; Thanked: 6 times; Followed by:5 members

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.

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

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]