According to the question, the four-digit number m must have the digits of rstu, since *m* = (3r)(5s)(7t)(11u).

If *n* = (25)(*m*)
*n* = (52)(3r)(5s)(7t)(11u)
*n* = (3r)(5s+2)(7t)(11u)

n is also a four digit number, so we can use the *n* value to identify the digits of n:

thousands = r, hundreds = s + 2, tens = t, units = u.

All of the digits of n and m are identical except for the hundreds digits. The hundreds digits of n is two more than that of m, so n – m = 200.

How do we know that the 2 digits equals 200 just becuase n is two more than that of m?

When you need to show exponents, use this symbol: ^

This problem should read:

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

So, first, m's four digits (in order) are rstu, yes, and n = (5^2)(3^r)(5^s)(7^t)(11^u) = (3^r)[5^(s+2)](7^t)(11^u)

so n's 4 digits (in order) are r(s+2)tu

next, n - m. The thousands, tens, and units digits of the 2 numbers are identical. The hundreds digit of n is 2 higher than the hundreds digit of m. What can we conclude?

Try some real numbers if you're not sure how this works in the abstract.

m = 1234
n = 1434
difference = 200

m = 4621
n = 4821
difference = 200

m = 3978
n = what, I can't do this! If I add 2 to 9, I get 11, but I'm supposed to have just a single digit in that spot... so there's a limit to what I can randomly pick to test the theory.

Okay, so why is it the case that n is always 200 larger than m?

Because n is always large by 2 digits in the hundreds place. The value of any digit in a number is always equal to the digit multiplied by the place value. In this case, the digit is 2 and the place value is 100. 2*100 = 200.

_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT

But it's about exponent, why aren't we calculating power of? 5^s+2 - 5^s

what makes you think we should?

_________________
Tim Sanders
Manhattan GMAT Instructor