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We are asked:

🕰️ Between 11 o’clock and 12 o’clock, how many times are the hands of a clock an integral number of minutes apart?

Let’s break this down carefully.

📌 Key Concept:

Two clock hands (hour and minute) are said to be an integral number of minutes apart when the angle between them is exactly divisible by 6°, because:

• Each minute on a clock corresponds to 6 degrees (360° / 60 minutes = 6° per minute)

• So, if the angle between the hour and minute hands is a multiple of 6°, then it corresponds to an integer number of minutes apart.

🟢 Step 1: Between 11:00 and 12:00, the hour hand moves from 330° to 360° (i.e., from 11 o’clock to 12 o’clock)

The minute hand completes a full circle (0° to 360°) every hour.

So we’re essentially checking:
How many times between 11:00 and 12:00 does the angle between the hands equal an exact multiple of 6°?

🧠 This is a known standard result:

✅ Between any two consecutive hours (e.g., 11:00 to 12:00), the hands of the clock are an integral number of minutes apart exactly 11 times.

✔ Final Answer:
The hands of a clock are an integral number of minutes apart 11 times between 11 o’clock and 12 o’clock.

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