Question:medium

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

Updated On: Jan 15, 2026

  • 3

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  • 5

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The Correct Option is C

Solution and Explanation

1. Given

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

The initial average of \( A, B, C \) is 13. Therefore, \( A + B + C = 3 \times 13 = 39 \).

2. Condition After Adding n

When \( n \) is added, there are four numbers: \( A, B, C, n \). The new average is \( \frac{A + B + C + n}{4} \).

This new average must be an odd integer. Let the odd integer be \( 2k + 1 \), where \( k \) is an integer.

So, \( \frac{39 + n}{4} = 2k + 1 \). Multiplying by 4 gives \( 39 + n = 4(2k + 1) = 8k + 4 \).

Rearranging for \( n \), we get \( n = 8k + 4 - 39 = 8k - 35 \).

3. Find Minimum Natural Number n

We require \( n \) to be a natural number, meaning \( n>0 \). We test integer values of \( k \) to find the smallest \( n>0 \).

  • For \( k = 0 \): \( n = 8(0) - 35 = -35 \) (Not a natural number)
  • For \( k = 1 \): \( n = 8(1) - 35 = -27 \) (Not a natural number)
  • For \( k = 2 \): \( n = 8(2) - 35 = -19 \) (Not a natural number)
  • For \( k = 3 \): \( n = 8(3) - 35 = -11 \) (Not a natural number)
  • For \( k = 4 \): \( n = 8(4) - 35 = -3 \) (Not a natural number)
  • For \( k = 5 \): \( n = 8(5) - 35 = 40 - 35 = 5 \) (A natural number)

The smallest natural number \( n \) that satisfies the condition is 5.

4. Final Answer

Correct Option: C. 5

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