Question

Prof. Bee noticed something peculiar while entering the quiz marks of his five students into a spreadsheet. The spreadsheet was programmed to calculate the average after each score was entered. Prof. Bee entered the marks in a random order and noticed that after each mark was entered, the average was always an integer. In ascending order, the marks of the students were 71, 76, 80, 82 and 91. What were the fourth and fifth marks that Prof. Bee entered?

• A. 71 and 82
• B. 71 and 76
• C. 71 and 80
• D. 76 and 80
• E. 91 and 80

Hint:

In stepwise average problems, check divisibility conditions systematically (by 2, 3, 4, etc.). Start with parity checks first, then eliminate impossible cases.

Answer 1

The Correct Option is C

Step 1: Key condition.
At every step, the average of the numbers entered must be an integer. - After 2 entries → sum must be divisible by 2. - After 3 entries → sum must be divisible by 3. - After 4 entries → sum must be divisible by 4. - After 5 entries → sum must be divisible by 5.

Step 2: Analyze parity (even/odd).
Marks: $\{71, 76, 80, 82, 91\}$. - Odd numbers: 71, 91 - Even numbers: 76, 80, 82 To make the first two divisible by 2, they must both be odd or both even.

Step 3: Try first pair (odd + odd).
Take first two as 71 and 91 → sum = 162, average = 81 (integer ✔). Now add third mark: 71+91+? must be divisible by 3. But sum = 162 + even number = even, not divisible by 3. So this case fails.

Step 4: Try first pair (even + even).
Choose two evens first (76, 80) or (76, 82) or (80, 82). Case (76, 80): sum = 156, average = 78 (✔). Now add third mark. Try 82 → sum = 238, average = 79.33 (not integer). Try 91 → sum = 247, average = 82.33 (not integer). Only possible is adding 82 → but fails. Case (76, 82): sum = 158, average = 79 (✔). Now add third mark. Try 80 → sum = 238, average not integer. Try 91 → sum = 249, divisible by 3 (✔), average = 83. So far sequence: (76, 82, 91).

Step 5: Add fourth mark.
Now sum = 249. Add 71 → total = 320. Divisible by 4 (✔), average = 80. Then add 80 last → total = 400. Divisible by 5 (✔), average = 80. So sequence works.

Step 6: Identify 4th and 5th marks.
Fourth = 71, Fifth = 80.

Final Answer: \[ \boxed{71 \; \text{and} \; 80} \]