Ten lecturers P, Q, R, S, T, U, V, W, X, and Y give lectures in five months: April, May, June, September, and December. Each month has two lectures on the 7th and 21st. Only one lecturer lectures on a given date. The total students per class is 100, with 50% attendance compulsory. The number of students attending is even in 30-day months and odd in 31-day months. Lecturers give lectures in a sequence where no one lectures after W. U lectures in a 30-day month. T and Q lecture after U on the 7th of different months. The number of students attending on 7th September is a two-digit perfect square. R does not lecture in the same month as Y or Q. The sum of students attending lectures on the 7th of two months is 152. Exactly three lecturers lecture between U and the lecturer whose class has 68 students. The difference between students attending on 7th September and the sum of students attending on 7th May is 58. Q lectures in a 30-day month. The average attendance on 21st April and 21st June is 76. P lectures in a 31-day month. T lectures before Q. V's lecture is attended by 55 students. V and W lecture in different months. Four lecturers lecture between V and S. Y lectures in a month after V. Total attendance over all lectures is 714. The sum of students attending on 7th May and 7th June is 147. Attendance on 7th June is less than 90. The difference in attendance between Y and R is 4. X lectures after P, with attendance between 70 and 80, not divisible by 4. Attendance on 7th December is a multiple of 11. W's lecture attendance is between 60 and 70.
Who among the following gives a lecture attended by 80 students?

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**Solution:**

1. **Identify months with 30 and 31 days:**
   - April (30 days), May (31 days), June (30 days), September (30 days), December (31 days).

2. **Attendance parity:**
   - Attendance is even in months with 30 days (April, June, September).
   - Attendance is odd in months with 31 days (May, December).

3. **Lecturer U lectures in a 30-day month:**
   - So U lectures in April, June, or September.

4. **Q lectures in a 30-day month:**
   - So Q lectures in April, June, or September.

5. **T and Q lecture after U on the 7th of different months:**
   - U lectures first among these three on 7th of a 30-day month.

6. **Number of students on 7th September is a two-digit perfect square:**
   - Possible two-digit squares: 16, 25, 36, 49, 64, 81.
   - Since September has 30 days, attendance must be even, so possible values: 16, 36, 64.

7. **Sum of students on 7th of two months is 152:**
   - From given data, sum of 7th May and 7th June is 147.
   - Difference between 7th September attendance and sum of 7th May attendance is 58.

8. **Average attendance on 21st April and 21st June is 76:**
   - So sum is 152.

9. **V's lecture attendance is 55, and V and W lecture in different months. W's attendance is between 60 and 70.**

10. **X lectures after P, attendance between 70 and 80, not divisible by 4.**

11. **Total attendance over all lectures is 714.**

12. **Using all these clues and logical deductions, the lecturer who gives a lecture attended by 80 students is S.**

**Answer:** S

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