For any four digit number, abcd, *abcd*= (3^a)(5^b)(7^c)(11^d). What

Question

For any four digit number, abcd,  *abcd*= (3^a)(5^b)(7^c)(11^d) . What is the value of (n – m) if m and n are four digit numbers for which  *m* = (3^r)(5^s)(7^t)(11^u)  and  *n* = (25)(*m*) ?

A. 2000
B. 200
C. 25
D. 20
E. 2

Bunuel · Accepted Answer

Given for four digit number,  abcd ,  *abcd*=3^a*5^b*7^c*11^d ;

From above as  *m*=3^r*5^s*7^t*11^u  then four digits of  m  are  rstu ;

Next,  *n*=25*\{*m*\}=5^2*(3^r*5^s*7^t*11^u)=3^r*5^{(s+2)}*7^t*11^u , hence four digits of  n  are  r(s+2)tu , note that  s+2  is hundreds digit of  n ;

You can notice that  n  has 2 more hundreds digits and other digits are the same, so  n  is 2 hundreds more than  m :  n-m=200 .

Answer: B.

Or represent four digits integer  rstu  as  1000r+100s+10t+u  and four digit integer  r(s+2)tu  as  1000r+100(s+2)+10t+u  -->  n-m=(1000r+100(s+2)+10t+u)-1000r+100s+10t+u=200 .

Answer: B.

KarishmaB · Answer

Also, if you find to difficult to grasp a question with many variables, try throwing in some values. It helps you handle the question.

abcd is a four digit number where a, b, c and d are the 4 digits.
*abcd*= (3^a)(5^b)(7^c)(11^d). The '**' act as an operator.

Given: *m* = (3^r)(5^s)(7^t)(11^u)
So m = rstu where r, s, t, and u are the 4 digits of m.
Say, r = 1 and s = 0, t = 0 and u = 0
m = 1000
Then *m* = 3

Now, 
*n* = (25)(*m*) = 25(3) = (3^1)(5^2)(7^0)(11^0)
n = 1200

n - m = 1200 - 1000 = 200

enigma123 · Answer

Sorry buddy - apologies if I am missing something. But how did you get

four digits of m as r-s-t-u;

Bunuel · Answer

For four digit integer  abcd  some function, denoted by **, defined as  *abcd*=3^a*5^b*7^c*11^d .

Now, as given that  *m*=3^r*5^s*7^t*11^u  then four digits of m are  rstu , the same way as above:  *rstu*=3^r*5^s*7^t*11^u

Hope it's clear.

enigma123 · Answer

Ahhh - so four digits are r, s, t and u. And not r minus s minus t minus u. That's where I got confused.

dayasantoshi · Answer

Hi  Bunuel

Can we arrive at the solution by the following approach ?

Given: *m* = 3^r*5^s*7^t*11^u
 *n* = 25 (*m*)

To Solve: n - m

Sol: Substituting for n , 
 n - m = 25 *m* - *m*
 = *m* (25-1)
 = *m* (24)
we know that, 24 = 3*2^3 and *m* = 3^r*5^s*7^t*11^u , does not have 2 value which implies the answer should have 2^3 as a factor.

1. 2000 = 5^3*2^4 - ( Only 2^3 is possible. as 24 has only 2^3 and *m* is not a factor of 2)
2. 200 = 5^2*2^3 - Correct
3. 25 = 5^2
4. 20 = 5 *2^2
5. 2 = 2

abrakadabra21 · Answer

n-m = 24m 
why 200 is not divisible by 24. What am i missing

Bunuel · Answer

How did you get that n - m = 24? We are given that *n* = (25)(*m*), not that n = 25m. Notice that both n and m are functions (*n* and *m*, not n and m).

sushforgmat · Answer

This is a good question as it is tricky with many variables

energetics · Answer

+1 for plugging in... once we get that the 5s are the hundreds place, just make every variable 0 so each place value is now 1. 
So, we get n-m = 1(1+2)11 - 1111 = 200

pravalithota · Answer

n=25m
n-m=25m=m=24m= 8 * 3m
ie., m is a multiple of 8
2000 and 200 are multiples of 8 
Now,
2000=250/3=(25*10)/3=(5^3)*2*(3^-1)
But m is not a factor of 2 as defined in the question.
200=25/3=(5^2)*(3^-1)
Therefore,
answer is B

KarishmaB · Answer

Responding to a pm: The tricky thing about this question is to understand that we are using an operator on a 4 digit number abcd. For a 4 digit number abcd, *abcd*= (3^a)(5^b)(7^c)(11^d) similar to f(abcd) = (3^a)(5^b)(7^c)(11^d) Then given *m* = (3^r)(5^s)(7^t)(11^u) = f(rstu) it means m = rstu *n* = (25)(*m*) = 25 * (3^r)(5^s)(7^t)(11^u) = (3^r)(5^{s+2})(7^t)(11^u) so n = r(s+2)tu i.e. the hundreds digit of n is 2 more than that of m. Now assume m = 1111, then n = 1311 and the difference between them is simply 200. Answer (B) Check this video on Functions: https://youtu.be/ik5XijHCOAI and this post: https://anaprep.com/algebra-functions/

piyushnagre99 · Answer

Hi  Bunuel  ,

Do these questions still appear in the GMAT exam?

Bunuel · Answer

This question tests functions, number properties, and exponents, all of which are still part of the GMAT Focus Edition. So yes, such a question can still appear on the GMAT.

For any four digit number, abcd, abcd= (3^a)(5^b)(7^c)(11^d). What

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