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CAT 2022 Slot 1 — QA Question 19

Change in AverageEasy

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:

Answer & solution

  • 5

  • B

    1

  • C

    3

  • D

    4

Solution

Easy

Convert the averages to sums. The new four-number average being an odd integer forces 39+n39+n to be a multiple of 88 (since 4×odd4\times\text{odd}). Find the smallest natural nn that does this.

1

Sum of the three integers:

sum=3×13=39\begin{aligned} &\text{sum}=3\times 13=39 \end{aligned}
2

Set the new average equal to an odd integer 2k−12k-1 and clear the fraction:

39+n4=2k−1⇒ 39+n=8k−4⇒ 43+n=8k\begin{aligned} &\frac{39+n}{4}=2k-1\\ &\Rightarrow\ 39+n=8k-4\\ &\Rightarrow\ 43+n=8k \end{aligned}
3

Make 43+n43+n a multiple of 88. Since 43=8×5+343=8\times5+3, we need n≡5(mod8)n\equiv 5\pmod 8. The smallest natural value is:

n=5 ⇒ 43+5=48=8×6 ✓\begin{aligned} &n=5\ \Rightarrow\ 43+5=48=8\times 6\ \checkmark \end{aligned}

Minimum n=5n=\mathbf{5}. Option (a).

CAT 2022 Slot 1 QA Q19: The average of three integers is 13. When a natural number n is included, the average of these four integers r — Solution | TheCATExam