CAT 2022 Slot 1 · Quant · Question 7

Three integers averaging 13 have a total of 3\times 13=39.

Adding n makes the total 39+n spread over four numbers.

The new average is \frac{39+n}{4}, and this must be an odd integer.

Writing that odd integer as k, 39+n=4k with k odd.

So n=4k-39, and n must be a natural number, hence at least 1.

Trying k=9 gives n=36-39=-3, which is not natural.

Trying the next odd value, k=11 gives n=44-39=5, which is natural.

No odd k between these exists, so 5 is the smallest workable value.

Checking: the four numbers total 39+5=44, averaging 11 — an odd integer, as required.

The minimum is 5 — the third option.

Shortcut. Test the options directly against \frac{39+n}{4}: n=1 gives 10, n=3 gives 10.5, n=4 gives 10.75, and n=5 gives 11. Only the last is an odd integer.