Question

The average of three integers is 13. When a natural number n is included, the average of these four integers remains an odd integer. The minimum possible value of n is

• A. 3
• B. 4
• C. 5
• D. 1

Answer 1

Answer: (C)

5

Answer 2

The correct answer is C :5
Let's denote the three integers as A, B, and C, and the natural number as n. We are given that the average of these three integers is 13, so we have:
$\frac{(A+B+C)}{3}=13$
A+B+C = 39
Now, we want to find the minimum value of n such that when it is included, the average becomes an odd integer. When we add n to the sum and
divide by 4 (since we now have four integers), we want the result to be an odd integer. An odd integer can be represented as 2k+1, where k is an
integer.
So, we have:
$\frac{(A+B+C+n)}{4}=2k+1$
Substitute the value of A + B + C from the first equation:
$\frac{(39+n)}{4}=2k+1$
Now, we want to find the minimum value of n that satisfies this equation. Let's try different values of k and see what n values work:
For k = 0:
$\frac{(39 + n)}{4}=1$
39+n = 4
n=-35 (which is not a natural number)
For k = 1:
$\frac{(39+n)}{4}=3$
39+n=12
n=-27 (which is not a natural number)
For k = 2:
$\frac{(39 + n)}{4} = 5$
39+n=20
n=-19 (which is not a natural number)
For k = 3:
$\frac{(39 + n)}{4}=7$
39+n=28
n=-11 (which is not a natural number)
For k = 4:
$\frac{(39 + n)}{4} = 9$
39+n=36
n=-3 (which is not a natural number)
For k = 5:
$\frac{(39+n)}{4}=11$
39+n=44
n=5
Since we're looking for a natural number n, the minimum possible value of n is 5. When n is 5, the average of the four integers becomes an odd integer (11).