CAT 2022

Slot 1

QA

Question & Solution

1. Concept Used

• Topic: Averages & Divisibility Conditions
• Formula: $$ \text{New Average} = \frac{\text{Sum of original integers} + n}{4} $$

2. Calculation

We are told that the average of three integers is 13, so their total sum is ( 3 \times 13 = 39 ).

When the natural number ( n ) is added, the new average of four numbers becomes ( \frac{39 + n}{4} ), and this must be an odd integer.

For ( \frac{39 + n}{4} ) to be an integer at all, ( (39 + n) ) must be divisible by 4. Since ( 39 \equiv 3 \pmod{4} ), we need ( n \equiv 1 \pmod{4} ), meaning ( n ) can be ( 1, 5, 9, 13, \ldots )

Now, for the average to be odd, ( \frac{39 + n}{4} ) must be an odd integer. Let's check the smallest candidates:

• If ( n = 1 ): ( \frac{39 + 1}{4} = \frac{40}{4} = 10 ) → Even ❌
• If ( n = 5 ): ( \frac{39 + 5}{4} = \frac{44}{4} = 11 ) → Odd ✅

So the minimum value of ( n ) that makes the average an odd integer is ( n = 5 ), giving a new average of 11.

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3. Solution

Answer = Option C ✅

The minimum possible value of ( n ) is 5.