M7MBA wrote: ↑

Fri Feb 19, 2021 8:49 am

2019-09-21_1528.png

Let \(n\) and \(k\) be positive integers with \(k \le n.\) From an \(n \times n\) array of dots, a \(k \times k\) array of dots is selected. The figure above shows two examples where the selected \(k \times k\) array is enclosed in a square. How many pairs \((n, k)\) are possible so that exactly \(48\) of the dots in the \(n \times n\) array are NOT in the selected \(k \times k\) array?

A. 1
B. 2
C. 3
D. 4
E. 5

Answer: C

Source: Official Guide

In general, an n x n array will have a total of n² dots
Likewise, an k x k array will have a total of k² dots
So, n² - k² = the number of dots in the n × n array that are NOT in the k × k array

If there are 48 dots in the n × n array but NOT in the k × k array, we can write: n² - k² = 48
This means we're looking for integer pairs (n, k) such that k ≤ n that meet the following condition: n² - k² = 48
This task is made easier if we recognize that n² - k² is a difference of squares, which can be easily factored

Factoring a difference of squares: [m]x^2 - y^2 = (x+y)(x-y)[/m]
This means the equation above can be factored as follows: (n + k)(n - k) = 48
There are five pairs of integers that multiply together to get 48 (48 & 1, 24 & 2, 16 & 3, 12 & 4, 8 & 6)

So, there are five possible cases to consider:
case i) (n + k) = 48 and (n - k) = 1
case ii) (n + k) = 24 and (n - k) = 2
case iii) (n + k) = 16 and (n - k) = 3
case iv) (n + k) = 12 and (n - k) = 4
case v) (n + k) = 8 and (n - k) = 6

Important: Before we select answer choice E, we must recognize that two of 5 cases do not meet the given conditions.
Take case i for example.
If n + k = 48 and n - k = 1, then we can add the two equations to get: 2n = 49, which means n = 24.5 and k = 23.5
Since the question tells us that n and k are positive integers, case i doesn't work

We'll use the same procedure to test the remaining 4 cases
case ii) n + k = 24 and n - k = 2. Add the equations to get 2n = 26, which means n = 13 and k = 11. So, case ii works
case iii) n + k = 16 and n - k = 3. Add the equations to get 2n = 19, which means n = 9.5. So, case iii doesn't work
case iv) n + k = 12 and n - k = 4. Add the equations to get 2n = 16, which means n = 8 and k = 4. So, case iv works
case v) n + k = 8 and n - k = 6. Add the equations to get 2n = 14, which means n = 7 and k = 1. So, case v works

Answer: C