Arithmetic

Mean, Median & Mode

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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

$3$

$5$

$4$

$1$

✅ Correct Option: 2

We have three integers with an average of 13. When we add a natural number n, the average of all four integers must remain an odd integer. We need to find the minimum possible value of n.

Average of 3 integers = 13

Sum of 3 integers = 13 × 3 = 39

When natural number n is included:

Sum of 4 integers = 39 + n

Average of 4 integers = $\frac{39 + n}{4}$

The problem states this average must be an odd integer.

For $\frac{39 + n}{4}$ to be an odd integer, two conditions must be met:

$(39 + n)$ must be divisible by 4 (so the result is an integer)

The quotient $\frac{39 + n}{4}$ must be odd

Let's check what remainder 39 gives when divided by 4:

39 ÷ 4 = 9 remainder 3

So 39 ≡ 3 (mod 4)

For $(39 + n)$ to be divisible by 4:

39 + n ≡ 0 (mod 4)

3 + n ≡ 0 (mod 4)

n ≡ 1 (mod 4)

This means n can be: 1, 5, 9, 13, 17, ...

Now we test the smallest possible values:

When n = 1: $\frac{39 + 1}{4} = \frac{40}{4} = 10$ (even)

When n = 5: $\frac{39 + 5}{4} = \frac{44}{4} = 11$ (odd)

With n = 5:

Sum of 4 integers = 39 + 5 = 44

Average = $\frac{44}{4} = 11$ (which is indeed an odd integer)

Therefore, the minimum possible value of n is 5.

When dealing with averages that must satisfy specific conditions (like being odd), we always:

Set up the divisibility condition first

Then check the additional constraint (odd/even)

Test systematically from the smallest possible values

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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