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*)?
- 2000
- 200
- 25
- 20
- 2
“For any four digit number, abcd, *abcd*= (3^a)(5^b)(7^c)(11^d). ”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.
Solution and Explanation:
Approach Solution 1:
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: (s+2) is hundreds digits of n.
Candidates can notice that n has 2 more hundreds digits and other digits are the same, so n is 2 hundreds more than m:
Hence, n-m = 200
Correct Answer: B
Approach Solution 2:
Also, if the candidates find it too difficult to grasp a question with many variables, they can try throwing in some values. It will help them to 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
Correct Answer: B
Approach Solution 3:
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.
Correct Answer: B
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