For a three-digit number xyz, where x, y, and z are the : Problem Solving (PS)

testgmat

Joined: 15 Jan 2008

Last visit: 07 Mar 2008

Posts: 5

Own Kudos:

2

Posts: 5

Kudos: 2

29 Jan 2008, 13:53

Kudos

Bookmarks

Incognito,

Could you please detail your work?

Thanks

incognito1

JCLEONES

For a three-digit number xyz, where x, y, and z are the digits of the number,
f(xyz)=5^x 2^y 3^z . If f(abc)=3*f(def), what is the value of abc-def ?
(A) 1
(B) 2
(C) 3
(D) 9
(E) 27

Since f(abc) = 3*f(def), I would assume that f = c - 1 from the function above.

The answer should be (A).

maratikus

Joined: 01 Jan 2008

Last visit: 22 Jul 2010

Posts: 257

Own Kudos:

340

Given Kudos: 1

Posts: 257

Kudos: 340

29 Jan 2008, 13:57

Kudos

Bookmarks

JCLEONES

For a three-digit number xyz, where x, y, and z are the digits of the number,
f(xyz)=5^x 2^y 3^z . If f(abc)=3*f(def), what is the value of abc-def ?
(A) 1
(B) 2
(C) 3
(D) 9
(E) 27

f(abc)/f(def)=5^(a-d) 2^(b-e) 3^(c-f) = 3^1 -> a = d, b = e, c = f+1 -> A is the answer

incognito1

Joined: 26 Jan 2008

Last visit: 11 Dec 2016

Posts: 160

Own Kudos:

266

[3]

Given Kudos: 16

Posts: 160

Kudos: 266

[3]

29 Jan 2008, 14:03

2

Kudos

1

Bookmarks

testgmat

Incognito,

Could you please detail your work?

Thanks

incognito1

JCLEONES

For a three-digit number xyz, where x, y, and z are the digits of the number,
f(xyz)=5^x 2^y 3^z . If f(abc)=3*f(def), what is the value of abc-def ?
(A) 1
(B) 2
(C) 3
(D) 9
(E) 27

Since f(abc) = 3*f(def), I would assume that f = c - 1 from the function above.

The answer should be (A).

Sure. The function consists purely of powers of 5, 2 and 3. None of which are multiples of each other.

Since f(abc) = 3*f(def), and since the function is entirely made up of multiples of 5, 2 and 3, it would imply that the only difference is a power of 3. This would mean that a = d (since there is no multiple of 5 from the functions output), b = e (no multiple of 2 in the functions output) and c = f + 1 (since there is a multiple of 3 in the two functions output). Since abc and def are 3-digit numbers, and c = f + 1, the difference between abc and def would be 1. Hope that helps!

eschn3am

Joined: 12 Jul 2007

Last visit: 03 Apr 2017

Posts: 396

Own Kudos:

573

Q50 V45

Posts: 396

Kudos: 573

29 Jan 2008, 21:32

Kudos

Bookmarks

JCLEONES

For a three-digit number xyz, where x, y, and z are the digits of the number,
f(xyz)=5^x 2^y 3^z . If f(abc)=3*f(def), what is the value of abc-def ?
(A) 1
(B) 2
(C) 3
(D) 9
(E) 27

f(abc)=3*f(def) so the only difference is one more multiple of 3 in f(abc). The only way for this to happens is if z is one more. z is the units digit so it's total value is 1 more.

Answer A

kyatin

Joined: 29 Jan 2007

Last visit: 16 Oct 2016

Posts: 250

Own Kudos:

158

[2]

Location: Earth

Q50 V40

Posts: 250

Kudos: 158

[2]

29 Jan 2008, 21:56

1

Kudos

1

Bookmarks

I went totally tangent on this one.

thought abc is a*b*c and not a number abc

and wasnt sure why every one was thinking its too obvious.

lesson learnt! :shock:

Bunuel

Math Expert

Joined: 02 Sep 2009

Last visit: 11 Jul 2025

Posts: 102,636

Own Kudos:

740,669

[4]

Given Kudos: 98,172

Products:

Expert

Expert reply

Posts: 102,636

Kudos: 740,669

[4]

12 Jul 2014, 03:44

1

Kudos

3

Bookmarks

JCLEONES

For a three-digit number xyz, where x, y, and z are the digits of the number, f(xyz)=5^x 2^y 3^z . If f(abc)=3*f(def), what is the value of abc-def ?

(A) 1
(B) 2
(C) 3
(D) 9
(E) 27

mvictor

Board of Directors

Joined: 17 Jul 2014

Last visit: 14 Jul 2021

Posts: 2,126

Own Kudos:

1,249

[1]

Given Kudos: 236

Location: United States (IL)

Concentration: Finance, Economics

Schools: Stanford '20 (WD)

GMAT 1: 650 Q49 V30 Verified score

GPA: 3.92

WE:General Management (Transportation)

Products:

Schools: Stanford '20 (WD)

GMAT 1: 650 Q49 V30 Verified score

Posts: 2,126

Kudos: 1,249

[1]

06 Nov 2015, 21:57

1

Kudos

Bookmarks

oh yeah..tricky one
f(def) = 5^d*2*e*3^f
a is the same multiplied by 3
we have same prime factorization
abc = def+1
abc-def = 1

Nunuboy1994

Joined: 12 Nov 2016

Last visit: 24 Apr 2019

Posts: 559

Own Kudos:

122

Given Kudos: 167

Location: United States

Schools: Yale '18

GMAT 1: 650 Q43 V37 Verified score

GRE 1: Q157 V158

GPA: 2.66

Schools: Yale '18

GMAT 1: 650 Q43 V37 Verified score

GRE 1: Q157 V158

Posts: 559

Kudos: 122

23 Feb 2017, 20:10

Kudos

Bookmarks

https://gmatclub.com/forum/the-function ... 00847.html

The answer is "A" - VeritasPrepKarishma gives an incredibly clear explanation in a similar problem linked above.

nightvision

Joined: 04 Mar 2018

Last visit: 11 Sep 2020

Posts: 20

Own Kudos:

9

[1]

Given Kudos: 34

GPA: 3.5

Products:

Posts: 20

Kudos: 9

[1]

19 May 2018, 13:38

Kudos

1

Bookmarks

testgmat

Incognito,

Could you please detail your work?

Thanks

incognito1

JCLEONES

For a three-digit number xyz, where x, y, and z are the digits of the number,
f(xyz)=5^x 2^y 3^z . If f(abc)=3*f(def), what is the value of abc-def ?
(A) 1
(B) 2
(C) 3
(D) 9
(E) 27

Since f(abc) = 3*f(def), I would assume that f = c - 1 from the function above.

The answer should be (A).

f(abc) = 5^a * 2^b *3^c
f(def) = 5^d * 2^e * 3^f

since, f(abc) = 3*f(def)
5^a * 2^b *3^c = 5^d * 2^e * 3^(f+1)

so, a=d
b=e
c=f+1

so, for example, if abcis 123, then def is 124,

hence difference is 1.

CEdward

Joined: 11 Aug 2020

Last visit: 14 Apr 2022

Posts: 1,212

Own Kudos:

247

Given Kudos: 332

Posts: 1,212

Kudos: 247

01 Dec 2020, 17:18

Kudos

Bookmarks

rgajare14

JCLEONES

For a three-digit number xyz, where x, y, and z are the digits of the number,
f(xyz)=5^x 2^y 3^z . If f(abc)=3*f(def), what is the value of abc-def ?
(A) 1
(B) 2
(C) 3
(D) 9
(E) 27

This is how I solved.
f(abc) = 3 * f(def)
So, 5^a*2^b*3^c = 3*[5^d*2^e*3^f]
So, (5^a*2^b*3^c)/(5^d*2^e*3^f) = 3
So, 5^(a-d)*2^(b-e)*3^(c-f) = 3^1
So,5^(a-d)*2^(b-e)*3^(c-f) = 3^1*5^0*2^0
S0, a-d =0, b-e =0 and c -f =1.
So, a =d, b = e and c = f +1
So abc - def is equal to abc - abf
And since c = f + 1, differrence is 1
Answer is A

My issue was even getting to this step:
So, 5^a*2^b*3^c = 3*[5^d*2^e*3^f]

How are we to conclude that abc = def = xyz?

I never even got off the ground on this one.

MathRevolution

Math Revolution GMAT Instructor

Joined: 16 Aug 2015

Last visit: 27 Sep 2022

Posts: 10,078

Own Kudos:

18,736

[2]

Given Kudos: 4

GMAT 1: 760 Q51 V42 Verified score

GPA: 3.82

Expert

Expert reply

GMAT 1: 760 Q51 V42 Verified score

Posts: 10,078

Kudos: 18,736

[2]

02 Dec 2020, 09:53

2

Kudos

Bookmarks

f(xyz) = \(5^x 2^y 3^z\)

=> f(abc)=\(5^a 2^b 3^c\)

=> f(def)=\(5^d 2^e 3^f\)

=> f(abc)=\(5^a 2^b 3^c\) = 3 * f(def)=\(5^d 2^e 3^f\)

=> \(\frac{[5^a 2^b 3^c]}{ [5^d 2^e 3^f]}\) = 3

=> \(5^{a - d} 2^{b - e} 3^{e - f}\) = 3

=> \(5^{a - d} 2^{b - e} 3^{e - f} = 5^0 2^0 3^1\)

Comparing the bases:

=> a - d = 0
=> b - e = 0
=> c - f = 1

Adding all:

=> (a + b + c) - (d + e + f) = 0 + 0 + 1

Answer A

Kinshook

Major Poster

Joined: 03 Jun 2019

Last visit: 12 Jul 2025

Posts: 5,698

Own Kudos:

5,212

[1]

Given Kudos: 161

Location: India

Schools: HBS '26 CBS '26 Wharton '26

GMAT 1: 690 Q50 V34 Verified score

WE:Engineering (Transportation)

Products:

Schools: HBS '26 CBS '26 Wharton '26

GMAT 1: 690 Q50 V34 Verified score

Posts: 5,698

Kudos: 5,212

[1]

21 Dec 2021, 18:19

1

Kudos

Bookmarks

Given: For a three-digit number xyz, where x, y, and z are the digits of the number, f(xyz)=5^x 2^y 3^z .
Asked: If f(abc)=3*f(def), what is the value of abc-def ?

f(abc)=3*f(def)
5^a 2^b 3^c = 3* 5^d 2^e 3^f = 5^d 2^e 3^(f+1)
b = e;
a = d;
c = f+1;
f = c -1
abc-def = abc - ab(c-1) = 1

IMO A

RastogiSarthak99

Joined: 20 Mar 2019

Last visit: 10 Aug 2024

Posts: 142

Own Kudos:

21

[1]

Given Kudos: 282

Location: India

Posts: 142

Kudos: 21

[1]

17 Aug 2022, 08:09

1

Kudos

Bookmarks

I think there is an easier way to look at this:

5^a 2^b 3^c = 3 * 5^d 2^e 3^f

--> 5^a 2^b 3^c = 5^d 2^e 3^f+1

a = d
b = e

c = f+ 1
c-f = 1

abc
(-)def
---------
001

A

gmatmonster805

Joined: 14 Aug 2023

Last visit: 21 Jun 2024

Posts: 2

Own Kudos:

1

[1]

Given Kudos: 22

Posts: 2

Kudos: 1

[1]

10 Jun 2024, 11:27

1

Kudos

Bookmarks

Answer

\[ f(abc) = 3 \cdot f(def) \]

So,

\[ 5^a \cdot 2^b \cdot 3^c = 3 \cdot (5^d \cdot 2^e \cdot 3^f) \]

Dividing both sides by \( 5^d \cdot 2^e \cdot 3^f \), we get:

\[ \frac{5^a \cdot 2^b \cdot 3^c}{5^d \cdot 2^e \cdot 3^f} = 3 \]

Which simplifies to:

\[ 5^{a-d} \cdot 2^{b-e} \cdot 3^{c-f} = 3^1 \]

Equating the exponents on both sides, we have:

\[ 5^{a-d} \cdot 2^{b-e} \cdot 3^{c-f} = 3^1 \cdot 5^0 \cdot 2^0 \]

Therefore,

\[ a - d = 0 \]
\[ b - e = 0 \]
\[ c - f = 1 \]

Thus,

\[ a = d \]
\[ b = e \]
\[ c = f + 1 \]

So, \( abc - def \) is equal to \( abc - abf \).

Since \( c = f + 1 \), the difference is 1.

Answer: \( \boxed{A} \)

bumpbot

Non-Human User

Joined: 09 Sep 2013

Last visit: 04 Jan 2021

Posts: 37,376

Own Kudos:

1,010

Posts: 37,376

Kudos: 1,010

13 Jun 2025, 09:09

Kudos

Bookmarks

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

For a three-digit number xyz, where x, y, and z are the

Prep Toolkit

My Rewards

GMAT Problem Solving (PS) Questions

For a three-digit number xyz, where x, y, and z are the

Prep Toolkit

My Rewards