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

The Correct Option is C

1. Given

Let the three integers be \( A, B, C \), and let the natural number to be added be \( n \).

Given average of the three numbers: \[ \frac{A + B + C}{3} = 13 \Rightarrow A + B + C = 39 \]

2. Condition After Adding n

New average becomes: \[ \frac{A + B + C + n}{4} \] which must be an odd integer.

Let an odd integer be represented as \( 2k + 1 \). Then: \[ \frac{39 + n}{4} = 2k + 1 \] Multiply both sides by 4: \[ 39 + n = 4(2k + 1) = 8k + 4 \Rightarrow n = 8k + 4 - 39 = 8k - 35 \]

3. Find Minimum Natural Number n

We want \( n > 0 \). Try increasing integer values of \( k \):

• For \( k = 0 \): \( n = -35 \) ❌
• For \( k = 1 \): \( n = -27 \) ❌
• For \( k = 2 \): \( n = -19 \) ❌
• For \( k = 3 \): \( n = -11 \) ❌
• For \( k = 4 \): \( n = -3 \) ❌
• For \( k = 5 \): \( n = 5 \) ✅

So, the smallest natural number \( n \) such that the new average is odd is: \[ \boxed{5} \]

4. Final Answer

Correct Option: C. 5

Answer 2

1. Given

Average of three numbers = \( 13 \)

Total sum of the three numbers: \[ 3 \times 13 = 39 \]

2. Adding a Fourth Number (n)

New average with four numbers: \[ \frac{39 + n}{4} \] It is given that this average must be an odd integer.

3. Finding the Minimum Natural Number

The smallest odd integer greater than \(\frac{39}{4} = 9.75\) is 11. 
Set: \[ \frac{39 + n}{4} = 11 \] Multiply both sides: \[ 39 + n = 44 \Rightarrow n = 44 - 39 = 5 \]

4. Final Answer

Minimum value of n is: \(\boxed{5}\) 
Correct Option: (C)