**Introduction:**
Vineet follows a cleaning routine for his car that includes vacuuming, washing, and polishing. He vacuum cleans his car every fourth day, washes it every fifth day, and polishes it every eighth day. He performed all three activities on 1st June. We need to determine when he will be doing all three activities on the same day again.

**Analyzing the Frequency of Activities:**
To find the next occurrence of Vineet performing all three activities on the same day, we need to analyze the frequencies of each activity individually.

1. Vacuum Cleaning: Vineet vacuum cleans his car every fourth day.
- The days when Vineet will vacuum clean his car are 1st June, 5th June, 9th June, 13th June, and so on.

2. Washing: Vineet washes his car every fifth day.
- The days when Vineet will wash his car are 1st June, 6th June, 11th June, 16th June, and so on.

3. Polishing: Vineet polishes his car every eighth day.
- The days when Vineet will polish his car are 1st June, 9th June, 17th June, 25th June, and so on.

**Finding the Common Day:**
We need to find the day when Vineet will be performing all three activities on the same day. To do this, we need to find the least common multiple (LCM) of the frequencies of each activity.

The LCM of 4, 5, and 8 is 40.

**Determining the Next Occurrence:**
Since the LCM of the frequencies is 40, Vineet will be performing all three activities on the same day every 40 days.

The next occurrence will be 40 days after 1st June, which is 11th July.

Therefore, Vineet will be vacuum cleaning, washing, and polishing his car on the same day again on 11th July.

**Conclusion:**
Vineet follows a cleaning routine for his car, performing vacuum cleaning every fourth day, washing every fifth day, and polishing every eighth day. By analyzing the frequencies of each activity and finding the least common multiple (LCM), we determined that Vineet will be performing all three activities on the same day again on 11th July, which is 40 days after 1st June.

9 june